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Criticism of the Foundations of the Relativity Theory
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S. N. Arteha
CRITICISM OF THE FOUNDATIONS OF THE RELATIVITY THEORY

The present book is devoted to systematic criticism of the fundamentals of the relativity theory (RT). The main attention is given to the new logical contradictions of RT, since presence of such contradictions brings ”to zero” the value of any theory. Many disputable and contradictory points of this theory and its corollaries are considered in detail in the book. The lack of logical and physical grounding for fundamental concepts in the special and general relativity theory, such as time, space, the relativity of simultaneity etc., is demonstrated. A critical analysis of experiments that resulted in the generation and establishment of relativity theory is presented in the book. The detailed criticism of dynamical SRT concepts is also given in the book. The inconsistency and groundlessness in a seemingly ”working” section of the relativity theory – the relativistic dynamics – is shown.
The given book can be of interest to students, post-graduates, teachers, scientists and all mans, that independently meditate on fundamental physical problems.

Contents

Preface

5

1 Kinematics of special relativity theory

11

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Relativistic time . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Relativity of simultaneity . . . . . . . . . . . . . . . . . . 33

1.4 The Lorentz transformations . . . . . . . . . . . . . . . . 38

1.5 Paradoxes of lengths shortening . . . . . . . . . . . . . . . 41

1.6 The relativistic law for velocity addition . . . . . . . . . . 51

1.7 Additional criticism of relativistic kinematics . . . . . . . 60

1.8 Conclusions to Chapter 1 . . . . . . . . . . . . . . . . . . 71

2 The basis of the general relativity theory

73

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.2 Criticism of the basis of the general relativity theory . . . 74

2.3 Criticism of the relativistic cosmology . . . . . . . . . . . 99

2.4 Conclusions to Chapter 2 . . . . . . . . . . . . . . . . . . 104

3 Experimental foundations of the relativity theory

106

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.2 Criticism of the relativistic interpretation of series of ex-

periments . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.3 Conclusions to Chapter 3 . . . . . . . . . . . . . . . . . . 134

4 Dynamics of the special relativity theory

135

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3

4

CONTENTS

4.2 Notions of relativistic dynamics . . . . . . . . . . . . . . . 137 4.3 Criticism of the conventional interpretation of relativistic
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.4 Conclusions to Chapter 4 . . . . . . . . . . . . . . . . . . 183
Appendixes

A On possible frequency parametrization

185

B Possible mechanism of the frequency dependence

194

C Remarks on some hypotheses

200

Afterword

206

Bibliography

212

Preface
This book is dedicated to my kind honest wise parents
Though the technology achievements have been quite impressive in the elapsed century, the achievements of science should be recognized to be much more modest (contrary to ”circumscientiﬁc” advertising). All these achievements can be attributed, most likely, to eﬀorts of the experimenters, engineers and inventors, rather than to ”breakthroughs” in the theoretical physics. The ”value” of ”post factum arguments” is well-known. Besides, it is desirable to evaluate substantially the ”losses” from similar ”breakthroughs” of the theorists. The major ”loss” of the past century is the loss of unity and interdependence in physics as a whole, i.e. the unity in the scientiﬁc ideology and in the approach to various areas of physics. The modern physics obviously represents by itself a ”raglish blanket”, which is tried to be used for covering boundless ”heaps” in separate investigations and unbound facts. Contrary to the artiﬁcially maintained judgement, that the modern physics rests upon some well-veriﬁed fundamental theories, too frequently the ad hoc hypotheses appear (for a certain particular phenomenon), as well as science-like adjustments of calculations to the ”required result”, similarly to students’ peeping at an a priori known answer to the task. The predictive force of fundamental theories in applications occurs to be close to zero (contrary to allegations of ”showman from science”). This relates, ﬁrst of all, to the special relativity theory (SRT): all practically veriﬁable ”its” results were obtained either prior to developing
5

6

PREFACE

this theory or without using its ideas, and only afterwards, by the efforts of ”SRT accumulators”, these results have been ”attributed” to achievements of this theory.
It may seem that the relativity theory (RT) has been ﬁrmly integrated into the modern physics, so that there is no need to ”dig” in its basement, but it would be better to ﬁnish building ”the upper stages of a structure”. One can only ”stuﬀ the bumps” when criticizing RT (recall the resolution of the Presidium of the USSR Academy of Sciences, that equated the RT criticism to the invention of the Perpetuum Mobile). The solid scientiﬁc journals are ready to consider both the hypotheses, which can not be veriﬁed in the nearest billion of years, and those hypotheses, which can never be veriﬁed. However, anything but every scientiﬁc journal undertakes to consider the principal issues of RT. It would seem the situation has to be just opposite. Because RT is being teached not only in high schools, but also in a primary school, at arising even slightest doubts all issues should be seriously and thoroughly discussed by the scientiﬁc community (in order ”not to spoil young hearts”).
However, there exists (not numerous but very active and of high rank) part of scientiﬁc elite that behaves a strangely encoded manner. These scientists can seriously and condescendingly discuss ”yellow elephants with pink tails” (superheavy particles inside the Moon that remained obligatory after Big Bang, or analogous fantasies), but an attempt to discuss the relativity theory leads to such active centralized acts, as if their underclothes would be taken oﬀ and some ”birth-mark” would be discovered. Possibly, they received the ”urgent order to inveigh” without reading. But any criticism, even most odious, can have some core of sense, which is able to improve their own theory.
RT claims to be not simply a theory (for example, as one of computational methods as applied to the theory of electromagnetism), but the ﬁrst principle, even the ”super-supreme” principle capable of canceling any other veriﬁed principles and concepts: of space, time, conservation laws, etc. Therefore, RT should be ready for more careful logical and experimental veriﬁcations. As it will be shown in this book, RT does not withstand logical veriﬁcation.

PREFACE

7

Figuratively speaking, SRT is an example of what is called an ”impossible construction” (like the ”impossible cube” from the book cover, etc.), where each element is non-contradictive locally, but the complete construction is a contradiction. SRT does not contain local mathematical errors, but as soon as we say that letter t means the real time, then we immediately extend the construction, and contradictions will be revealed. A similar situation takes place with spatial characteristics, etc.
We have been learned for a long time to think, that we are able to live with paradoxes, though the primary ”paradoxes” have been reduced by relativists rather truthfully to some conventional ”strangenesses”. In fact, however, every sane man understands that, if a real logical contradiction is present in the theory, then it is necessary to choose between the logic, on which all science is founded, and this particular theory. The choice can obviously not be made in favor of this particular theory. Just for this reason the given book begins with logical contradictions of RT, and the basic attention is given to logical problems here.
Any physical theory describing a real phenomenon can be experimentally veriﬁed according to the ”yes - no” principle. RT is also supported by the approach: ”what is experimentally unveriﬁable – it does not exist”. Since RT must transfer to the classical physics at low velocities (for example, for the kinematics), and the classical result is unique (it does not depend on the observation system), the relativists often try to prove the absence of RT contradictions by reducing the paradoxes to a unique result, which coincides with classical one. Thereby, this is a recognition of the experimental indetectability of kinematic RT eﬀects and, hence, of their actual absence (that is, of the primary Lorentz’s viewpoint on the auxiliary character of the relativistic quantities introduced). Various theorists try to ”explain” many disputable RT points in a completely diﬀerent manner: everybody is allowed to think-over the nonexistent details of the ”dress of a bare king” by himself. This fact is an indirect sign of the theory ambiguity as well. The relativists try to magnify the signiﬁcance of their theory by co-ordinating with it as many theories as possible, including those in absolutely non-relativistic areas. The artiﬁcial character of such a globalistic ”web” of interdependencies

8

PREFACE

is obvious.
The relativity theory (as a ﬁeld of activity) is defended, except the relativists, also by mathematicians, who forget that physics possesses its own laws. First, the conﬁrmability of some ﬁnal conclusions does not prove truth of the theory (as well as the validity of the Fermat theorem in no way implies the correctness of all ”proofs” presented for 350 years; or, the existence of crystal spheres does not follow from the visible planet and stars motion). Second, even in mathematics there exist the conditions, which can hardly be expressed in formulas and, thus, complicate searching for solutions (as, for example, the condition: to ﬁnd the solutions in natural numbers). In physics this fact is expressed by the notion termed ”the physical sense of quantities”. Third, whereas mathematics can study any objects (both really existing and unreal ones), physics deals only with searching for interrelations between really measurable physical quantities. Certainly, a real physical quantity can either be decomposed into the combination of some functions or substituted into some complex function, and then we can ”invent” the sense of these combinations. But this is nothing more than the scholar mathematical exercises on substitutions, which have nothing in common with physics irrespective of their degree of complication.
We shall leave for conscience of ”showman from science” their intention to deceive or to be deceived (to their personal interests) and shall try to impartially analyze some doubtful aspects of RT.
Note that during the RT life time the papers have repeatedly appeared, which contained some paradoxes and criticism of relativistic experiments; the attempts were undertaken to correct RT and to revive the theory of ether. However, the criticism of RT had only partial character, as a rule, and aﬀected only separate aspects of this theory. The current of the criticism and its quality was considerably increased in the end of the last century only (the article and book titles from the bibliography speak for themselves).
It should be recognized that, as against the criticism, there exists the professional fundamental apologetics of RT [3,17,19,26,30,31,33-35,3741]. Therefore, the main purpose of the author was to present a successive, systematic criticism of RT just resting upon a ﬁne apologetics of

PREFACE

9

this theory. Following to the ”conventional private tradition”, the basic part of the given book was tested in international scientiﬁc journals (GALILEAN ELECTRODYNAMICS, SPACETIME & SUBSTANCE). As a result this task has been fulﬁlled step-by-step beginning with the works [48-55], in which the author considered in detail the RT underlying experiments, the baseline kinematic concepts of the special relativity theory and of the general relativity theory, the notions of relativistic dynamics and some consequences of relativistic dynamics. The critical works contain, virtually, no papers on the relativistic dynamics. This fact was one of the main incentives for writing this book.
The present book represents by itself some generalization of published papers from the single standpoint. (Besides, for readers the logical subtleties can always be better grasped in own native language.) To see the most complete ”picture of nonsense” we shall, whenever possible, try to discuss each doubtful point of relativity theory irrespective of remaining ones. However, due to the limited scope, the book does not contain the citing from textbooks. Therefore, it is presupposed some reader’s knowledge of relativity theory. Besides, often the book considers both the conventional interpretations of relativity theory and possible ”relativistic alternatives”. This is made to prevent the temptation of rescue of relativity theory with other relativistic choices in disputable points. ”Monster” is dead for a long time, and it is not worth to revive it – this is the author’s opinion.
It is rather diﬃcult to choose the successive logic of presentation: for any problem there arises the desire for presentation of all attendant nuances in the same place of the book, but it is impossible. The author believe that if a reader can read to the end, majority of impromptu questions and doubts will be consecutively elucidated. The structure of the book is the following. Chapter 1 critically analyzes relativistic notions, like time, space, and many other aspects of relativistic kinematics. Chapter 2 presents the criticism of the basis for general relativity theory (GRT) and for relativistic cosmology. The experimental substantiation of RT will be criticized in Chapter 3. In so doing we shall not consider in detail the experiments pertinent only to electromagnetism or various particular hypotheses of ether (this theme is huge in itself). Instead, we

10

PREFACE

shall analyze exclusively some general experiments aﬀecting the essence of RT kinematics and dynamics. Chapter 4 contains the criticism of notions of special relativity theory (SRT), results and interpretations of relativistic dynamics. Conclusions are made for each chapter. Some particular hypotheses are considered in Appendixes.

Chapter 1
Kinematics of special relativity theory
1.1 Introduction
Traditionally, standard SRT textbooks begin with a description of the allegedly then existing crisis of physics and experiments that preceded the generation and establishment of SRT. However, there exists the opinion [38] that SRT was originated as a pure theoretical ”breakthrough” having no need of any experimental substantiation. The author does not agree with such a view, for physics is destined primarily to explain the really existing world and to ﬁnd interrelations between observed (measurable) physical quantities. Nevertheless, we begin the book with the theoretical consideration of relativistic kinematics, not with the analysis of experiments. The matter is that several theories can try to interpret the same observed phenomenon in quite diﬀerent ways (such is and will indeed the case for physics). However, it is common practice to abandon the theory manifesting logical contradictions. The history of physics demonstrates repeated changes of conventional interpretations for many phenomena. And it is not to be believed that the elapsed century was the last one for these changes.
In textbooks on general and theoretical physics, and in the popular scientiﬁc literature, there exists almost advertising support of spe-
11

12

CHAPTER 1. SRT KINEMATICS

cial relativity theory (SRT). This is expressed in headings like: ”on the Practical Importance of SRT”, ”on the Uniqueness and Foundation of all Mathematical Derivations and Corollaries from SRT”, ”on the Simplicity and Elegance of all SRT Results”, ”on Full Conﬁrmation of SRT by Experiments”, ”on the Absence of Logical Inconsistencies in SRT”, etc. But if we keep aside issues of particle dynamics (they will be discussed in Chapter 4), and consider only kinematic notions, then the ”Practical Signiﬁcance of SRT” will be obviously zero. The uniqueness and theoretical foundation of SRT can also be attacked [58,65,102,111]. In papers [48-50,52] a series of logical contradictions, related to the basic notions of space, time, and relativity of simultaneity, was analyzed in detail and the complete lack of logical grounding for SRT was proved. Also, the complete lack of experimental grounding for SRT was shown (these issues will be considered in Chapter 3 of the book); and as a demonstration that SRT is not uniquely implied by anything, the possibility of a frequency parameterization of all SRT results was described (although such a parameterization was not the main purpose of the cited work; it will be presented in Appendixes as a particular hypothesis).
In this Chapter, criticism of kinematic notions in SRT will be presented in detail, and attention will be given to some apparent errors from textbooks. All these circumstances force us to return to classical notions of space and time, as advanced by Newton. He formulated these notions in Mathematical Principles of Natural Philosophy as a brilliant generalization of works of precursors (including ancient Greeks). Relativists aspired to destroy the former conceptions at any cost (carping, basically, at the word ”absolute”) and to allege ”something new and great”. They could present no deﬁnitions for notions of time, space and motion, but only manipulated with the mentioned words. Therefore, though brief comments on Newton’s classical notions [28] ought to be given in Introduction.
Proceeding from practical demands of natural science, Newton understood that any creature is ”excellently familiar” with the mentioned notions and practically uses” theirs (for example, insects that are incapable of abstract thinking in opinion of people). So, these notions are the basic ones, i.e. they cannot be deﬁned through anything. Then, it is

1.1 INTRODUCTION

13

possible to give only an enumeration of ”things” that will be meant by these notions or will be used in practice and to separate the abstraction that will be implied for idealized mathematical calculations. Because of this, Newton clearly separated absolute, true, mathematical time or duration (all these words simply are synonyms in this case!) from relative, seeming or ordinary time. Thus, time means the mathematical comparison between duration of the process under investigation and duration of the standard process. In classical physics the possibility of introducing the universal time has not been directly connected with the obvious restriction on the speed of signal transmission. More likely obtaining the universal time was connected with the possibility to recalculate it from local times with reasonable exactness. In perfect analogy to this, Newton separated the absolute space notion from the relative one, distinguished absolute and relative place, and distinguished between absolute and relative motions. If the search of relationships of cause and eﬀect is believed to be one of the goal of sciences, then the important positive moment of the classical approach consists in a separation of an object under investigation from the rest of the Universe. For example, in the overwhelming majority of cases ”the motion of observer’s eyes” does not exert any noticeable inﬂuence on a concrete proceeding process and, so all the more, on the rest of the Universe. Certainly, there exist ”seeming eﬀects”, but to concentrate just upon the process under study, they can be eliminated by the graduating of devices, recalculations etc.. The classical kinematic notions was actually introduced by Newton just for the determination of registration points and standards independent of the process under investigation. This founds the grounds for the common description of diﬀerent phenomena, for the joining of various ﬁelds of knowledge and for the simpliﬁcation of the description. Also classical notions intuitively coincide with ones given to us in sensations: it is stupid not use they – it equals ”to try to go by ears”. A centuries-old development of sciences (from ancient Greeks) shows that the classical kinematic concepts lead neither to internal logical contradictions nor to discrepancy with experiments.
Now we shall pass to ”the things, created by relativists” in this ﬁeld, and consider logical contradictions in the fundamental notions of ”space”

14

Chapter 1. SRT KINEMATICS

and ”time” in SRT. We begin with the conception of time.

1.2 Relativistic time
Now we remind, how the erroneousness of RT kinematic concepts can be proved most easily. For the ”yes-no”-type results only one of diﬀerent evidences of two observers could be true. Therefore, at least one of moving observers would be wrong in mutually exclusive judgements. However, the situation can always be made symmetrical with respect to the third resting observer. Then his evidences will coincide with the classical (checked for v = 0) result, and in this case the evidences of both ﬁrst and second observers should transfer to this result. However, since both the ﬁrst and second observer moves relative to the third one, all three their evidences will be diﬀerent. Owing to situation symmetry, both the ﬁrst and second observer occurs to be wrong in his judgements, and only the third, resting observer describes the true (classical) result. Exactly in this manner the inconsistency of the concept of time (the time is irreversible!) was proved in the modiﬁed paradox of the twins [48,51], as well as the inconsistency of the ”relativity of simultaneity” concept [50]. (Note that the space-time diagram [33] does not change the physics of even conventional paradox of the twins: all additional aging of Earth’s inhabitant arises suddenly (!), when the motion of an astronaut changes at the far point and is only geometrically expressed as the change of lines of simultaneity).
We begin the detailed analysis of relativity theory with a modiﬁed ”twins paradox”.
The modiﬁed twins paradox
We would remind that in classical physics results are obtained by one observer can be used by any other observer (including investigators not participating in experiments). In such a case, our goal is to formulate some symmetric setting of a problem with results which are evident from the common sense. Relativists renounce the common sense permanently! Therefore, to prove the lack of contradictions and observability of rela-

1.2 RELATIVISTIC TIME
v

15
v

A

A

1

O
Figure 1.1: The modiﬁed twins paradox.

B

B

1

tivistic eﬀects, they must consider diﬀerent results from the viewpoint of diﬀerent observers and compare all results between themselves. However, for some reason they do not aspire to the Truth in this question. But few investigators, who carried out such analysis, either ascertained the lack of observable relativistic eﬀects for two-observer schemes (and announced it), or discovered the presence of contradictions for a larger number of observers (the most honest-minded peoples even passed to the camp of critics of RT).
Let two colonies of Earth’s inhabitants A and B be at some large distance from each other (Fig. 1.1). A beacon O is at the middle of this distance. It sends a signal (the light sphere), and when it reaches both colonies (simultaneously), each launches a spacecraft piloted by one twin. The laws of acceleration (to reach a large equal speeds) are chosen equal in advance. At the time each twin passes the beacon, at a high relative velocity, each will believe that his counterpart should be younger. But this is impossible, since they can photograph themselves at this instant and write their age on the back side of a picture (or even exchange pictures by the digital method). It is nonsense, if wrinkles will appear on a pictured face of any astronaut during the deceleration of another one. Besides, it is unknown beforehand if one of astronauts will wish to move with acceleration in order to turn around and catch up to the other one.
This paradox can be more reinforced and be formulated as a paradox of coevals – people born in the same year. (In SRT it is declared a change of time course rather than a transfer of initial time, as the time zone on the Earth, for example.) Let now the spacecrafts be launched with families of astronauts. Babies are born on each spacecraft just after

16

Chapter 1. SRT KINEMATICS

accelerations became equal to zero (accelerations were chosen equal in advance). And these babies are chosen for a comparison of age. All previous history of motion (to the points A1 and B1 respectively) does not exist for theirs. The observers at the points A1 and B1 can conﬁrm the fact of the baby birth. The babies diﬀer in that they moved relative to each other at speed 2v all the time. They travelled the equal distances |OA1| = |OB1| to the meeting. This is just the pure experiment to compare the time duration and to verify SRT. Let, for example, the ﬂight of the baby 1 with the constant speed v take place for a time 15 years. Then, from the SRT viewpoint, the ﬁrst baby will reason in the following manner: the second baby moved relative to me with a large velocity for all my life (15 years); therefore, his age must be less than mine. Besides, if he will count out the age of the second baby starting from the moment of the receipt of signal from B1, then he will believe that he will see infant in arms at the meeting. But the second baby will reason about the ﬁrst baby in the same manner. However, owing to full symmetry of the motion, the result is obvious: the age of both astronauts are the same (this fact will be conﬁrmed by the observer at the beacon).
Recall the explanation of the classical paradox of twins (one an astronaut and one an Earth’s inhabitant). These twins have ”unequal in rights,” since only one of them accelerates (it is just this person who was declared to be younger than the other one). But before acceleration each of the twins thought that the other one should be younger! And, in fact, if one twin is accelerated, then the other grows old faster. (Maybe, it makes sense to prohibit accelerating astronauts and sportsmen in order that everybody around could grow old to a less extent?). Even the ”explanation” of the classical twins paradox certainly contains some contradictions. First, everything could have been done symmetrically; the astronauts can take photographs before and after accelerations and even exchange pictures at the center (Can wrinkles appear on photos?!). Second, the explanation cannot lie in the acceleration. We see again at Fig. 1.1 (the modiﬁed twins paradox): even with initial equal accelerations, the twins can ﬂy at the same high relative velocity for diﬀerent times (due to diﬀerent initial distance |AB|, for example). For exam-

1.2 RELATIVISTIC TIME

17

ple, we choose these accelerations to be equal to the acceleration of free fall on the Earth. Then, the driving at high relativistic speed requires about one year (but all the distances can be chozen much more: 100 or 1000 light years). It is obvious that neither ”accelerated ageing” nor ”accelerated rejuvenation” can occur during this year (we can remember the equivalence of accelerated systems and systems in gravitational ﬁeld from the general relativity theory: just now we have conditions which are analogous to the usual Earth conditions!). It then occurs that accelerations the same in magnitude and in time of action at the same distances |AA1| and |BB1| may cause diﬀerent aging – depending on the time of previous motion (100 or 1000 years) at constant relative velocity (due to time slowing from SRT), i.e. there obtains a violation of causality. Further developing this idea, one can permanently change the sign of acceleration (< v >= 0), and an arbitrary additional aging will take place in this case (in such a case the SRT formulas for time slowing at a constant rate make no sense). Third, the accelerations and velocities may be diﬀerent for diﬀerent astronauts in the process of their motion, but their meeting can always be organized at the same point, and, by the opinion of each of astronauts, the age of the same object will be diﬀerent, that is nonsense.
Let us consider, for example, a modiﬁed paradox of ”n twins” (Fig. 1.2). Let them depart on ﬂights in diﬀerent directions from the same center O, so that all departure angles are diﬀerent in any pairs (we shall have an irregular n-gon). The schedule of velocities and accelerations is chosen the same beforehand (all spacecrafts are always ”situated” at some sphere with the center O). Because of vector character of quantities, all relative velocities and accelerations will be diﬀerent in pairs. By the opinion of some selected astronaut, each another astronaut must grow old to a diﬀerent age (and this takes place from the viewpoint of each astronaut), which is impossible (again all astronauts can photograph themselves before each acceleration and after it).
Attempts look naive when ”explanations” of diﬀerent versions of the classical twins paradox are ”made” with artiﬁcially fabricated auxiliary diagrams: relativists are again cunning and do not check results as a matter of contradictions from viewpoint of all observers (will somebody

18

Chapter 1. SRT KINEMATICS

O
Figure 1.2: The paradox of ”n twins”.
claim that the Lorentz transformations are insuﬃcient ones, but diagrams present something more thing? really?). Physics and mathematics are ”slightly” diﬀerent sciences to put it mildly. Possible, someone could be interested how pure geometric drawings (a rhombus, a parallelogram, a triangle etc.) can be turned or transformed to pseudoscientiﬁcally rescue the SRT. But these recommendations resemble the proud INSTRUCTIONS ”how one can scratch the right-hand ear with the left-hand heel, when this leg is twice wound round the neck, and can provoke the same sensations (they must be elucidated beforehand!) as the normal man (which satisﬁes his requirements in more natural manner). But even for such ”a state of aﬀairs”, the following fact is remarkable. In classical physics any logically consistent way leads to the same objective result (each observer can imagine reasoning of any other observer and even appropriates they). The matter is quite diﬀerent for SRT: it is ”necessary” to arbitrarily postulate some reasonings from absolutely single-type ones as false (i.e. there occur the ﬁtting the choice of a way to the classical result). The resulting theory is ”surprising”: ”here

1.2 RELATIVISTIC TIME K2

K’ 2’

1’

19 1

Figure 1.3: The time paradox viewed at t = 0.
we read, here we do not read, here we turn over a page by this manner, here we turn inside out by that manner”, and, as it is sung in the song: ”and in other things, the beautiful marchioness, a nice how-d’ye-do...”. It is concocted artfully.
The time paradox
Now we shall pass to the time paradox for moving systems. For ”resolving” it, the Lorentz transformations are often used: they allow one to put in correspondence to one time instant t the whole continuum of times t′. Note, that if we compare the time intervals, then the procedure of synchronizing the time reference point is unimportant. Let us have four clocks ((1, 2); (1′, 2′)), spaced similarly in pairs and synchronized in their own systems K and K′ (Fig. 1.3). The synchronization can, for example, be performed by an inﬁnitely remote source located on the axis perpendicular to the plane of all four clocks (it will be further outlined in the subsection on ”establishing the universal absolute time”). Then

20

Chapter 1. SRT KINEMATICS

K2

1

K’ 2’

1’

Figure 1.4: The time paradox viewed at t = t1.

for any intervals we have

∆t1 = ∆t2, ∆t′1 = ∆t′2

(1.1)

However, according to the Lorentz transformations formulas, from the point of view of observers in system K (near the clocks), at the time of coincidence of clocks we have (Fig. 1.4):

∆t′1 < ∆t1, ∆t′2 > ∆t2,

(1.2)

i. e. inequality (1.2) contradicts equality (1.1). A similar contradiction with (1.1) occurs if the inequalities are written from the point of view of observers in system K′ (near the clocks). Even the values of diﬀerences of time intervals will be diﬀerent. Thus, these four observers will not be able to agree among themselves, when they meet at one point and discuss the results. Where then is the objectiveness of science?

1.2 RELATIVISTIC TIME

21

The paradox of antipodes
The erroneousness of SRT is proved very simply by the whole life of mankind on the planet Earth. Let us consider the elementary logical contradiction of SRT – the paradox of antipodes. Two antipodes situated at the equator (for example, one person in Brazil, the other one – in Indonesia) diﬀer by the fact, that due to the Earth rotation they move relative to each other at constant speed at each time instant (Fig. 1.5). Therefore, despite the obvious symmetry of the problem, each of these persons should grow old or grow young relative to another one. Does the gravitation hinder? Let’s remove it and place each of our ”astronauts” into a cabin. Each person can determine the time on such a ”round robin” (as well as on the Earth) from the direction to the far star, which is motionless with respect to the round robin center, and from the period of intrinsic rotation of a round robin (a whirligig). The running of time will obviously be identical for both ”astronauts”. The time can be synchronized by the calculation technique knowing the period of revolution (all these problems are technological, rather than principal). Let’s increase the linear speed v → c for amplifying the eﬀect (for example, in order that according to SRT formulas the diﬀerence in time be ”running up” 100 years for one year). Does the centrifugal force (acceleration) hinder? Then we shall increase radius R of the round robin, so that v2/R → 0 (for example, in order that even for 100 years the overall eﬀect from such an acceleration be many orders of magnitude lower, than the existing accuracy of its measurement). In such a case none of experiments can distinguish the motion of antipodes from rectilinear one, i.e. the system non-inertialness cannot be experimentally detected throughout the test. It is worthless for relativists to ﬁght for the principal necessity of inertialness of the system. Recall that even in such the strict science as mathematics (in the justiﬁcation of the theory of real numbers, for example), it is used the notion of the number ε given beforehand, which can be chosen as small as one likes. In case discussed for the strict mathematical transition, the ratio of a centrifugal acceleration v2/R to the Earth’s centrifugal acceleration ac can be made less than any arbitrary value of ε at the expence of a large radius of a ”round robin” R (for instance, we can choose ε ∼ 10−10 or ε ∼ 10−100, whereas all SRT

22

Chapter 1. SRT KINEMATICS

V

s

V
Figure 1.5: The paradox of antipodes.

1.2 RELATIVISTIC TIME

23

experiments were made on the Earth with ε ∼ 1!). And, further, if you trust in the relativity (either according to SRT or according to Galileo – indiﬀerently, since we compare time durations), then you can transfer the motion of one of antipodes, in a parallel manner, closer to the other antipode and forget about the round robin model at all. Obviously, the reverse mental operation can always be performed for any two mutual opposite motions with the same speed as well. Namely, we can perform parallel transfer of one of trajectories to a great distance R → ∞ and ”bridge” the motions by some ”round robin”. So, will ”the patient be alive or dead” after some years? And who is more pleasant for you – the Brazilian or Indonesian? The full symmetry of the problem and full failure of SRT! Note, generally speaking, that the unique character of time cancels the principality of the issue of its synchronizing: the watch can, for example, be worn with yourself. Some doubts on ”near inertial” motions will be discussed below in Chapter 3. If some relativists will on principle try to connive (themselves and somebodies) at the possibility of such a transition to a large R, we can oﬀer to inscribe a regular n-gon into a circle of the large R (n ≥ 3; stationary observers are placed at all angles) and to consider pure rectilinear motions of spacecrafts with astronauts along the sides of the n-gon. Even the same loops for using the same ”earth” acceletations g (to gather the equal large speeds) can be joined to the angles of the n-gon in the identical manner. Obviously, all these inertial systems of the spacecrafts are absolutely identical for a stationary observer (at the center of the circle, for example). The course of time is the same for all spacecrafts in spite of diﬀerent relative motions of the spacecrafts. We can also draw the obvious symmetric scheme of ”a ﬂower type” with the possibility of the simultaneous start and ﬁnish of astronauts at the center of a circle (see Fig. 1.6).

Since we will compare the time course (but not time beginning), we

can use the equality of the time course for any two mutually resting

objects. Then, the model of a whirligig can be easily generalized to the

case of arbitrary (in directions and values) velocities of objects. This

is purely geometric trivial problem (Fig. 1.7). For example, let us have

t−Aw−A→o1

maontdio−Bn−sB→, 1w. hTichhe

are pictured in both velocities

Fig. 1.7 with the velocity vectors possess the same modulo v which

24

Chapter 1. SRT KINEMATICS

Figure 1.6: The symmetric model of ”a ﬂower”.

1.2 RELATIVISTIC TIME

25

B5

A3

A5

a

B3

C3

C5
d

C

C

1

A

A1

B1

D5

O

B A2

D
3

B4 B2

A4

Figure 1.7: The model of a whirligig for arbitrary planar motions.

tends to the speed of light v → c. Let us choose an arbitrary point O in the space. Furthermore, we draw a circle with the center at the point O and such a radius R that the centrifugal acceleration will be less than some preassigned small value ε1 (an existing accuracy of measurement of accelerations, for example): v2/R < ε1, i.e. R > v2/ε1. We draw the straight line AA2 which is perpendicular to the straight line AA1. Thereafter, we draw the line A3A4 passing through O and parallel to tvltAuhheAtleeoe2cmm.vitaooAylttuitivooenaenc|−Apt−Bwo−−oABri→→itn1h1−At|,−.3vo−wAe→fFel5oainoccwttbiteuhtyraaisclie−Alhncy−At,→miiso1wo.npetaiMoosrfniaamltklwhepiilnilstygthloimtnvh−Aeea−elAd→oaean1cnidataaynltpohd−Bag−ero3h−aBuc→alisls5rec.lptlherNtoerwoacwseenadsmdbularoretaetiwhoawnbttsihhtoohee-f motions are placed at the same circle and they cannot be distinguished from inertial motions with an existing accuracy. Due to obvious symmetry of the problem, the time course will be the same for these objects. For example, the time course can be measured with periodic ﬂashes, which occur at the center O of the circle. Now we take motion with

26

Chapter 1. SRT KINEMATICS

(atshhebeeesrotvelheualwtoteecitvtwtayalokuveeoe.cbttWhjoeercet−Crsma−C→d(aw1iku.eitsIhat|OpvisaeClrpo3aa|clrlieta=illeltesRrla−A|tn−−Co3−s−3Al−→−ACa→−5tA5→i|oa1/nn,|−Adbo−u3f−−CAt−→−C3−−5pCC→|→o)51.s)saewInsnsdiellstohmbsiotsoamviceneasa−Coe−lt3oh−Cwn→ege5r concentric arcs of circles A3a and C3d. These objects will remain at the same distance from each other along the radii of the circles. (Some big arcs are shown here for visualization only, i.e. all angular values are increased; in fact, all arcs will be very small and indistinguishable from rectilinear segments.) It is obvious that the time course for such objects will be the same. Time can again be ”measured oﬀ” by periodic ﬂashes from the center O (number of light spheres which are passed through the circle C3d is the same as for the circle A3a: the light spheres do not ”disappear, condense, add, or hide themselves” anywhere). We can also Atdargnaawgiennt,thiteahlecviroecblloejceictthtysrovwueigctthhortvh−Deel−o3−pcD→iot5iinewts iCt−Dh−33−Dta→hn5edasnaatdmae−Cn−a3y−Cb→ns5oelwaurteepopvilanaltcueeddr|a−Caw−t3−C→tthh5|ee. awsanimldl eb−De−c3i−trD→hcl5ee,s(aoamrnde−B.−,3−dTB→uh5euastn,odotnh−Ce−t3h−Cs→ey5me)xmwameetprpylreooovffedtmhotehtpiaortnostbhlweemitth,imtvheeeloctociiumtrieesesc−Aois−u3−rAi→nse5dependent on both the absolute value and the direction of the velocity of objects, but it is the same. Passage to the three-dimensional case is trivial. At the ﬁrst, we will transfer the beginning of one velocity vector to the beginning of the second velocity vector. Thereafter, we can draw a plane through these intersecting straight lines. In this plane we can carry out all previously described constructions. Thus, the time course is independent on any motions of inertial systems.

The universal absolute time
The notion of time is broader, than the dimensional factor in transformation laws, and bears much greater relation to the local irreversibility of processes. First, a single-valued ”binding” of time to the motion of a body does not take into account internal processes, which can be anisotropic, pass at various ”rates” and characterize the local irreversibility (each such rate is in diﬀerent manner added geometrically with the velocity of a body as a whole). Second, the binding of time

1.2 RELATIVISTIC TIME

27

v O
S
Figure 1.8: The interchange of signals of intrinsic time.
only to the velocity of transmission of electromagnetic interactions does not take into account other possible interactions (which can propagate in vacuum) and actually implies electromagnetic nature of all phenomena (the absolutisation of electromagnetic interactions). Later we shall consider, how the universal absolute time can be introduced.
When we introduce the notion of intrinsic time (actually, subjective time), the following methodological point seems important: We should not calculate intrinsic time of an alien object according to our own rules, but rather ”ask” this object itself. Consider the following experiment (Fig. 1.8): Let an observer be situated in the motionless system S at point O, where a beacon is installed. The beacon ﬂashes each second (as a result, the number of ﬂashes N equals the number of seconds passed at point O). Let an astronaut (in moving system S′) be launched from point O. Then, when moving away from point O the astronaut will perceive ﬂashes more rarely (at lower frequency), than before launching (in fact, beacon’s ”time slowing” takes place). But upon approaching to the beacon the astronaut will see the opposite, ﬂashes will occur more frequently than before launching (now we have beacon’s ”time speedup”). For v < c it is obvious that the astronaut can neither outstrip any ﬂashes, nor go around any of ﬂashes (light spheres). So regardless of his motion schedule and trajectory, upon returning to point O the astronaut will perceive equally N ﬂashes total, i.e. all ﬂashes, which have been emitted by a beacon. Therefore, each of these two observes

28

Chapter 1. SRT KINEMATICS

will conﬁrm that N seconds have passed at the beacon.
If the astronaut on board the spacecraft will also have a beacon and will signal about the number of seconds passed on his watch, then no disagreements will arise concerning astronaut’s time as well. The situation appears to be fully symmetrical (for the twins paradox, for example). When meeting at the same point, all light spheres will intersect opposite observers (their quantity can neither increase, no decrease). This number is equal to N - the number of seconds passed for both observers.
Consider now the problem of establishing the universal absolute time. (Of course, if we measure the time by beatings of our own heart, it will be subjective and will depend on the internal and external conditions). The attempt to introduce individual ”electromagnetic time” and to absolutize it – this is a return to the past. However, even at that time the people could synchronize time, despite miserable data transmission rate (by pigeon-post, for example), because they used a remote source of signals (the Sun or stars). Let us imagine the following mental experiment (Fig. 1.9). The remote source S, which lies on a middle perpendicular to segment AB, sends signals periodically (with period T ). At the time of signal arrival to point O, two recording devices (1 and 2) begin to move mirror-symmetrically (at velocities v and −v), while reﬂecting from A and B, with period of 2T . Velocity v can be arbitrary (we can choose the appropriate distance |AB|). In spite of the fact, that at each time instant the devices are moving relative to each other at speed 2v (except the reﬂection points), the signals will be received at the same time, namely, at the time of passing by point O (observer 3 can be placed at this point). The time, determined in such a manner, will be universal (at point O), i.e. the same for all three observers. In order to make the following step, we note that for deriving the transformation formulas in the SRT, it is suﬃcient to consider the relative motion along a single straight line (since the systems are inertial). By choosing the large distance |SO| we may assure that the time diﬀerence between signal arrival to point O and to points A and B be smaller than any pre-speciﬁed value. As a result, to the given accuracy the time will be the same for the whole chosen segment AB regardless of the velocities of motion of observers 1 and 2. Thus, the inﬁnitely remote source of signals, situated

1.2 RELATIVISTIC TIME

29

Ë

¹

©

Ê

¬ ¬ ¬ ¬ ¬

¬¬¬¬¬¬¬¬¬¬

½
¾¹
¿

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

Ç

Figure 1.9: An inﬁnitely remote source for establishing uniﬁed absolute time.

perpendicular to the direction of relative motion of systems, can serve as a watch counting the universal absolute time (which is the same regardless of the inertial system of reference). The question on the change in the observed direction of signal arrival will be presented below lest a temptation are going to arise in ”far-fetched” use of the aberration allegedly demonstrating the change in the wave front direction.
Additional remarks
The next methodological note is as follows: If the Einstein method is used for synchronization, the notion of time becomes limited. First, only one of two independent variables - spatial coordinates or time - remains independent, whereas the other is associated with the state of motion (subjectivism) and properties of light speed (but why is it not associated, for example, with the speed of sound or with the velocity of Earth, etc.?). Second, since the independent determination of spatial coordinates and time is required for determination of velocity, light speed itself becomes

30
S

Chapter 1. SRT KINEMATICS
S’

v

c

Figure 1.10: The light clock.

indeterminate quantity (immeasurable, postulated).
As relativists like to potter with idle inventions! One of such the ”Great” idle inventions of the relativity theory is a light clock (for 100 years anybody did not try to construct a pre-production model at all and will never try to make it!). And it is not because that it is impossible to create ideally ﬂat, ideally parallel, ideally reﬂecting mirrors. That is why that we cannot observe ”TICK-TOCK” sideways as it is described by the SRT. Such a clock ”works” to ﬁrst ”TICK” and ceases to be ”identical”, as a photon at the moment of ”TICK” registration should ﬁnally be reacted. Nevertheless, we will return ”to ours relativists”, which often use a ”light clock” for demonstrating the time slowing eﬀect [35] (Fig. 1.10). However, in exactly the same manner we can also consider a periodically reﬂecting particle (or a sound wave) at speed u ≪ c and obtain the arbitrary time slow-down τ0/ 1 − v2/u2. It is known, that the orthogonal velocity components can be described independently: the horizontal motion at velocity v relative to an instrument will in no way inﬂuence the vertical oscillations of a particle moving at former velocity u. The question on experimental veriﬁcations of the postulate of light speed constancy will be analyzed in Chapter 3.
The time slowdown in SRT is nothing else, but the apparent eﬀect. Remind that for a sound the duration of a hooting of trumpet ∆t also depends on the velocity of a receiver relative to a source (a trumpet),

1.2 RELATIVISTIC TIME

31

but nobody makes the conclusions on time slowdown from this fact. The fact is that observer’s ”decision” to move at any velocity is in no way bound causally with sound emitting processes (as well as with other processes in a trumpet). Let a singer be continuously singing a song in the resting atmosphere, and his twin brother be moving away from a singer at about the speed of sound vs : α1 ≡ v/vs ≈ 1, and then he will move toward a singer (with the same ratio α1). Though the song will be distorted, nobody had yet recorded more rapid aging of a singer. Let now we modulate with the same song the light in pursuit of the twin brother, who departed on a rocket at about the speed of light, but with the same numerical value α2 ≡ v/c = α1 ≈ 1. Now the twin brother will listen the same distorted song. Why the situation must change in this case, and the ”home seating” brother must grow old? And, if some living organism will be characterized by some certain radiation frequency, that distinguishes him from the dead organism, then, really, because of your motion (because of the Doppler eﬀect) you will ﬁrst certify the death of an organism, and then his resurrection? Or it is necessary to postulate the change of objective characteristics of an object, which is not bound with you causally?
Now we make some comments concerning Einstein’s time synchronization method. The transitivity of time synchronization by Einstein’s method takes place for the trivial case of three mutually resting points. If, however, the points (not lying on the same straight line) belong to the systems moving relative to each other in diﬀerent (not parallel) directions, then the synchronization procedure can become uncertain: for what time instant the watch can be considered to be synchronized? For the beginning of the procedure, for its termination or for an intermediate instant? Even for the points lying on the same straight line Einstein’s method rests upon a completely unveriﬁed (experimentally) concept of equality of the speed of light in one and in a directly opposite direction. Actually, the synchronization occurs to be either a half-done calculation procedure, or a multi-iterative process, because the synchronization is performed for two selected points only. These deﬁciencies are absent in the method of synchronization with a remote source disposed at a middle perpendicular [48]. It allows one to synchronize the time experimentally

32

Chapter 1. SRT KINEMATICS

(rather than computationally), without attracting additional hypotheses, to a prescribed accuracy throughout the given segment (even on a ﬂat section) at once.
Now we proceed to the time measurement units. Certainly, for a separate phenomenon within the framework of some mathematical model any customary quantity can be described in various measurement units and in various scales (both uniform and non-uniform, for example, in the logarithmic scale). This is basically determined both by the convenience of description for the given model, and, as in the case of generalization, by the possibility of using the same quantities for the other physical phenomena and mathematical models (the matching of various ﬁelds of physics). However, Taylor and Wheeler’s [33] sarcasm concerning the ”sacred units” is completely inadequate. Certainly, we can introduce the factor for converting the time into meters. But this factor is not obliged to be the speed of light: for example, it can be the velocity of a pedestrian. Both aforementioned velocities have, quite equally, no relation to acoustic, thermal phenomena, to hydrodynamics and to many other ﬁelds of physics. It is possible to express, generally, all quantities (such as mass, charge, etc.) in meters. However, all these ”various meters”:
1) can not be summed up,
2) are not interchangeable,
3) very rarely appear in some joint combinations and
4) the same combination is unsuitable for various phenomena. (For example, the interval has relation only to the law of light propagation in vacuum.) All quantities can be made pure numbers (and we must separately look after all these physical values). But in any case physics will not become mathematics. Physics does not study all illusory combinatorial ”worlds” of equations, but only that rather small amount of them, which is realized in the nature (the basic problems of physics are: what interrelations are realized in the nature, why and what are the consequences of this).

1.3. RELATIVITY OF SIMULTANEITY

33

1.3 Relativity of simultaneity

Now, after criticism of the fundamental concept of time for SRT, we continue the analysis of the logical basis of this theory and consider the subsidiary notion of the ”relativity of simultaneity”. Recall the mental experiment from SRT: a train A′B′ passes along a railroad at speed v. Suddenly, lightning strikes the railroad bed (C) just opposite to the train center C′ (at the moment of coincidence C = C′). Then, in the coordinate system centered on the moving train, the ﬂashes will simultaneously arrive at points A′ and B′, whereas for a motionless observer the ﬂashes will simultaneously arrive at points A and B (with the middle at point C); but up to this instant, points C and C′ (the middles of segments) will move to some distance from each other. But a similar situation is possible in classical physics as well, if we want to transmit information from points A′, B′, A, B to the new single point D (or, conversely, to these points A′, B′, A, B from D) at some ﬁnite speed v1 (in this case SRT and light speed constancy will be without any relevance).
We can suggest the following mechanical model (Fig. 1.11): Let four material points (without the force of gravity) fall at speed v1 in pairs over point C (close to the railroad bed) and over the train’s centre C′ which will arrive to the point C” near to point C at the moment of intercept of falling points. Let ideal reﬂectors (isosceles triangles with angle at a base α = π/4) be installed at point C and at train’s center. Then two particles, reﬂected over the railroad bed (at point C), will ﬂy to diﬀerent sides at speed v1, and simultaneously reach points A and B (in the classics |AB| = |A′B′|). This process will take time t = L/v1, where 2L is the length of the train. Two other particles, reﬂected over the train’s center C′, will move after reﬂection (relative to the railroad) at speeds v′ = v1 + (v/ tan α) = v1 + v (forwards) and v” = v1 − v (backwards). During the same time t the ﬁrst of these particles will traverse the path (forwards) L′ = v1t + vt, and, since the train traverses the path vt, the particle will reach point A′. Similarly, for the second particle L” = v1t − vt; hence, it reaches point B′. Thus, the event – the falling of points to the reﬂectors – will be

34

CHAPTER 1. SRT KINEMATICS

Figure 1.11: The mechanical model for the relativity of simultaneity.

1.3 RELATIVITY OF SIMULTANEITY

35

recorded at all four points simultaneously: both at points A and B (over the railroad bed), and at points A′ and B′ (over the train). It was the case when the points, falling over train, participated in its inertial movement. If the second pair of points falls (over the railroad bed) just over motionless point C” the triangular reﬂector at the train (only at it) should have the following corners at the basis: against the train movement - α3 = 0.5 arctan (v1/v), and in the direction of the train movement - α4 = π/2 − α3. In this case particles will ﬂy in parallel to the train and will reach its ends simultaneously (but not simultaneously with the second pair of particles!). If we want, that all four material points ”have ﬂown by” simultaneously corresponding points A′, B′, A, B, corners at the reﬂector basis (at the train) should be still reduced by corner arccos √ v1 (if to establish a ﬂat waveguide, the pair of particles
v2 +v12
over the train will ”not rise” too highly, and will move in parallel to the train). Apparently, mechanical analogues are possible for the most diﬀerent situations.
One can say that these two events are quite diﬀerent. But in the case of the light ﬂash, we have two diﬀerent events as well. Indeed, let the light ﬂash occur at the time the centers O and O′ of systems S and S′ moving relative to each other at v coincide. At some time instant t > 0, the light front will be on the sphere Σ relative to center O in system S and on the sphere Σ′ with center O′ in system S′ (which seems to be impossible). However, there is nothing surprising (i.e. contradicting classical physics) in this situation, because the observers in system S and S′ will record the same light to have diﬀerent frequencies ω and ω′ by virtue of the Doppler eﬀect. But in this case these are two identiﬁably diﬀerent events: the observers can always compare the results of measurements ω and ω′ upon meeting!
Consider now in detail the mental experiment allegedly ”demonstrating” the relativity of simultaneity: at the origins O and O′ of reference systems S and S′ that move relative to each other, a light ﬂash occurs at the time of their coincidence. According to SRT, during the time ∆t = t1 − t01 on the clock of system S, the light will pass the distance c(t1 − t01) from center O. For the same time ∆t = t2 − t02 on the clock of system S′, the same light will pass the distance c(t2 − t02) from center

36
S"
V
3

CHAPTER 1. SRT KINEMATICS

S

S’

V

2

1

Figure 1.12: The contradictions of the relativity of simultaneity.
O′. The time diﬀerence ∆t is not inﬂuenced by any adjustment of initial times, whether accomplished before the experiment, or after it by any method. For example, an inﬁnitely remote periodic source located perpendicular to the direction of motion can be used. It is possible to agree in advance about the ﬂashes, recorded on the clock of system S (for example, periodically each million years), and ”to organize” system S′ for one instant before the ﬂash occurs, selected in advance (the paradox of non-locality, associated with this, will be considered in Section 1.7).
Recall that the basic positive idea of SRT consisted in the ﬁniteness of the speed of interactions. The same idea is expressed by a short-range interaction theory, which reﬂects the ﬁeld approach (via the Maxwell equations); namely: a light wavefront moving from a source to a receiver passes sequentially through all intermediate points of space. It is just this property that comes in a conﬂict with the notion of relativity of simultaneity (Fig. 1.12). To prove it, we use two statements from the SRT about observers moving each relative other: 1) one and the same light ﬂash will reach two observers simultaneously despite the fact that the observers will spatially be separated by some distance during the light spreading; 2) kinematic formulas of the SRT (from textbooks) contain squares of velocities only. For example, let the ﬁrst observer in system S be moving towards the ﬂash source at slow speed v ∼ 104 m/s. Since the distance to the ﬂash point is large (say a million light years), then for one million years both observers will separate from each other to a large distance – about 2 · 1017 m. According to SRT formulas, the

1.3 RELATIVITY OF SIMULTANEITY

37

times of arrival of a signal will be the same for both observers. At what point of space did the ﬁrst observer ”pass” the light wavefront for the second observer? But what if he had held a mirror for the whole million years, and removed it one second before receiving a signal? In the second observer’s opinion, the signal was reﬂected by the ﬁrst observer somewhere ahead. But in this case what thing was reﬂected by the ﬁrst observer, if none of his instruments did still respond to a ﬂash? Similarly, a third observer can go away from the second one at the same velocity, but directed from the source. If the second observer held a mirror for a million of years except one second, would the third one see the light?
On the one hand, since the SRT formulas include the square of velocity only, the second observer will consider the time of signal reception by the ﬁrst and third observers to be the same. It can be agreed that when observers receive the signal under investigation, each of them will send his signal without delay. If second observer’s calculations are correct, then since the problem is symmetric, he must receive the signals from the ﬁrst and third observers simultaneously. On the other hand, according to Maxwell equations, the light propagates continuously, and the second observer will receive a signal from the ﬁrst one simultaneously with the event, when he himself will see the signal under investigation. In second observer’s opinion, at this time the light has still not reached the third observer. Thus, the second observer comes to a contradiction with himself: the ﬁrst calculations by SRT formulas contradict the second calculations by the Maxwell equations. Obviously, the observers will see the ﬂash sequentially, rather than simultaneously, since the spatial path of light is sequential: the source, the ﬁrst observer, then the second and, at last, the third observer.
We additionally note that even within the SRT framework the concept of the relativity of simultaneity is highly restricted: it is applicable to two separated events only (there are no intersecting original causes, no intersecting aftereﬀects, and, generally, we are not interested in any additional facts). Indeed, even for these selected points the light cones have intersections, to say nothing of all other points in space and time. In fact, we have continuous chains of causally bound (and unbound) events

38

CHAPTER 1. SRT KINEMATICS

occurring with multiple intersections through every point of space and time (not every reason, of course, results in a consequence at a speed of light). And all this real (diﬀerent in scale!) time grid is interdependent for the whole space. Therefore, in the general case we can not change (by choosing the frame of reference) the order of succession of even causally unbound events (in any case, this changing would be reﬂected somewhere).

1.4 The Lorentz transformations

Let us make some comments concerning the Lorentz transformations. One of the approaches to deriving these transformations uses the light sphere, which is visible in diﬀerent manner from two moving systems (the ﬂash took place at the time of coincidence of the centers of systems). Or, what is actually the same, this approach uses the concept of interval (displaying the same sphere). The solution of the system of equations

x2 + y2 + z2 = c2t2 x21 + y12 + z12 = c2t21

(1.3) (1.4)

represents simply the intersection of two surfaces and nothing more (Fig. 1.13). Under the condition of y = y1, z = z1 these ﬁgures will be the surfaces of a sphere and of an ellipsoid of rotation with the distance vt between the centers of the ﬁgures. However, this is actually the other problem – the problem on two ﬂashes: it is possible to ﬁnd the centers of the given ﬂashes for any time instant, i.e. to solve the reverse problem.
In the other approach to deriving the Lorentz transformations such a transformation is sought, which transfers equation (1.3) into equation (1.4). Obviously, for four variables such a transformation is not unique. First, the separate equating y = y1, z = z1 represents only one of possible hypotheses, as well as the requirement of linearity, mutual uniqueness, reversibility, etc. (An additional possibility of ω-parametrization is described in Appendixes.) Second, any transformation of light surfaces does not determinate at all the transformation of volumes (in which the non-electromagnetic physical processes may occur). For example, the

1.4 THE LORENTZ TRANSFORMATIONS

39

y

y

1

vt O O1

xx 1

Figure 1.13: The problem of two ﬂashes.

40
K

CHAPTER 1. SRT KINEMATICS
Z K’
v

Figure 1.14: The contradiction of a continuum of light spheres.

speed of sound does not depend on the motion of a source as well, but no global conclusions follow from this fact.
In any case, the Lorentz transformations in SRT physically describe two objects, rather than a single one. Otherwise it is easy to see a contradiction (Fig. 1.14). Let a light ﬂash occur. Let us separate, instead of a light sphere, one beam perpendicular to the mutual motion of systems K and K′ (and let the remaining light energy be absorbed inside the system). Let us block the path of a beam by installing the long mirror Z at a great distance from sphere’s center (along the line parallel to the line of mutual motion of systems). Then the observer situated at the center of system K will record the reﬂected signal after some time. Let the signal be completely absorbed. However, the other observer moving together with system K′ will catch a signal, also after some time, at the other point of space (let the signal be absorbed too). If we take a ”continuum” of systems with diﬀerent mutual velocities v, then the signal can be caught at any point of the straight line. Then where has the additional energy appeared from? May be this is SRT’s perpetuum mobile of the ﬁrst kind?
Note that if some mathematical equation is invariant relative the transformations of Lorentz type with some constant c′, it means only that among particular solutions of this equation there exist ”surfaces” of wave type which can propagate with the velocity c′. However, in this case even the given equation can have other particular solutions with other

1.5. PARADOXES OF LENGTHS SHORTENING

41

own invariant transformations, to say nothing of other mathematical equations, i.e. no overall mathematical conclusions do not follow from the fact of invariance. Only relativists try ”to blow the big soap-bubble” from the particular phenomenon.

1.5 Paradoxes of lengths shortening
Now we proceed to spatial concepts. Since all SRT conclusions follow from the invariance of an interval, then from the above-proved equality dt = dt′ and from (if we trust in it) relativistic equality c = constant we obtain dr = dr′, and so it is not necessary to consider the concept of space further at all. However, to form the most complete viewpoint we shall, whenever possible, consider in this book each disputable point irrespective of remaining ones.
The contraction of lengths in SRT can not reﬂect a real physical eﬀect, because various observers can see the same object in diﬀerent manner (the non-objectiveness). Besides, the transition from one frame of reference to another can proceed rather rapidly, and this transition would be reﬂected in the whole (even inﬁnite) Universe at once, which obviously contradicts the SRT-defended principle of ﬁnite rate of transmission of interactions and, hence the principle of causality. Therefore, a similar contraction is nothing more, than supplementary mathematical manipulations with quantities, some of which have no physical sense. The real physical mechanism can not be attracted to explaining the length contraction process in SRT, since the contraction should take place immediately at any velocity v = 0. In reality, however, it is clear, that in the acceleration process the object can not only be pushed, but also pulled behind yourself, and in such a case, instead of contraction, we would have stretching (experimentally detectable, by the way!). At slow constant acceleration this constant state of stretching would remain the same throughout the motion. Thus, the contraction will never begin.
Since SRT was created just as ”a game with Einstein’s light beams in absolutely empty space”, any pseudo-paradoxes with use of an electromagnetic ﬁeld (currents with contacts, lasers, light beams with mirrors etc.) can be easy resolved, and relativists slyly present them as allegedly

42

CHAPTER 1. SRT KINEMATICS

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À³
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¢¢ ¢¢ ¢¢¢¢¢¢ ¢¢ ¢¢¢¢ ¢¢ ¢¢¢¢¢¢ ¢¢ ¢¢¢¢ ¢¢ ¢¢¢¢ ¢¢ ¢¢ ¢¢¢¢ ¢¢ ¢¢¢¢ ¢¢ ¢¢¢¢¢¢

Figure 1.15: The paradox of a cross.

lack of contradictions in SRT. For this purpose they simply make substitution and instead of real paradoxes ”discuss” pseudo-paradoxes such invented or ”added” by them with any electric contacts, allegedly eﬀective explosions etc. So be attentive to such forgery! And now we proceed to some particular paradoxes of lengths shortening.
The paradox of a cross
Let a thin plate of large size lie on a solid plane. A small cross is cut out of the plate (Fig. 1.15). Let the length of this cross be much larger than its cross-beam width |AD| ≫ |BC|. Let the cross slide horizontally over the plate, so that in classical physics it would just occupy its niche and fall into it under the eﬀect of gravity. We choose the relative velocity of motion v such that, in accord with relativistic formulas, the length to be shortened two-fold (or even more). Note that the center of gravity of the cross (point o) lies also at the cross-beam center. Hence, vertical motions of the cross (falling down, or turning over its front end) is possible only if: 1) center o and the whole central line of a cross-beam (O′O”) are over empty space, and 2) none of points C, D, E, F has support. From the viewpoint of an observer on the cross, he will slide over a two-fold shortened niche, since either the cross-beam and one of ends, or both ends of the cross lean against the plate. The

1.5 PARADOXES OF LENGTHS SHORTENING

43

known trick with turning of a rod fails in this case (this problem will be considered below). However, from the viewpoint of an observer on the plate, the cross (which became two-fold shorter) will fall down into the niche. Thus, we have two diﬀerent events: does the downfall of the cross (a push against the plane) take place or not? and what will happen to the observer, who falls down into the niche (will he be crushed or not)? If he escapes, must he promptly begin to move with v (as well as the cross), or is he bound to be near the end A′H′ (or D′E′), because the cross became two-fold shorter? If someone very much wants to re-formulate this paradox as a paradox of existence, that (remembering the remark of the previous paragraph about relativistic ”electromagnetic forgeries”) a detonator should be under the plate, and a push-button contact could be closed under the plate in the center crosswise niche only with the center of gravity of the cross at its possible falling.

Additional paradoxes and ”strangenesses”
We can describe another paradox. Let the circle be cut oﬀ the plate and begin rotating around its center. Due to length shortening, an observer on the plate should see a clear space and the objects behind the plate. At the same time, the observer of the circle should see, how the plate runs over the circle. The noninertial character of the system does not matter, since the acceleration v2/R for v → c can be smaller than any prescribed value due to large R. The geometry of a circle will be considered in detail in Chapter 2 devoted to the general relativity theory. Similar contradictions demonstrate logical inconsistency of the habitual relativity theory (predictability – the foundation of science – is lost in this theory).
Note one more ”strange thing” (the paradox of distances). Since the shortening of lengths of objects is associated with properties of space itself, the distance to objects must also be shortened (regardless of whether we approach the object or move away from it!). Therefore, if the velocity of a rocket is high enough (v → c), we can not only look at distant stars, but also ”touch” them, because in our own reference system our own dimensions do not change. Besides, when ﬂying away from the Earth for a long time (the value of acceleration is not limited by SRT), we will

44

CHAPTER 1. SRT KINEMATICS

eventually be at the distance of just ”one meter” from it. At which time instant will the observer at this distance in ”one meter” see the reverse motion of the spacecraft (contrary to the action of rocket engines)?
The possibility of introducing the absolute time refutes logically paradoxical SRT conclusions about time slowing, relativity of simultaneity, and, besides, about distances shortening, because now the method of simultaneous measurement of distances does not depend on the motion of objects. Let an thin object (a contour portrait cut out a paper, for example) slide with an arbitrary velocity over the photographic ﬁlm, for example. If a momentary lighting is made by the inﬁnitely remote ﬂashlight, the length of the shadow photograph as well as the length of the object will the same. We can use an usual distant source (on a middle perpendicular to a plane) in the following case: the ﬂash front will reach the plane at a moment of ﬂight the middle perpendicular by the object (see p.1.7 below - about a ”seeming turn” of the wave front).
The distances to the objects are also contradictory for other reason. Even in motion at pedestrian speed, the distance to far galaxies must be noticeably contracted. However, the direction of such a contraction is indeterminate. If someone (moving) casts a look at galaxies, will he ﬂy away beyond Earth limits? Or, on the contrary, will he (moving) attract another galaxy by his glance? Any result is real mysticism!
A strange thing, related to length contraction in SRT, occurs with a belt-driven transmission (Fig. 1.16). From the viewpoint of the observers, on each of two free halves of a belt the cylindrical shafts should be transformed into ellipsoidal drums and then be turned as follows. The points of semimajor axes of ellipses, which are opposite to each observer, should approach each other (we obtain the non-objective description again). In SRT lengths of upper and lower half of the belt is found to be biassed, for instance. The contradiction takes place from the viewpoint of the third observer situated on a ﬁxed stand. On one hand, the shafts should approach each other. On the other hand, however, the ﬁxed bearing, which retains the spindles of shafts, should remain at the same place. But what is the thing, on which shafts’ spindles will be kept? So, whether the real space is contracted or not? What must be artiﬁcially postulated for urgent ”saving” SRT: various inserted spaces

1.5 PARADOXES OF LENGTHS SHORTENING

45

V

? ?

? ?

?

V

?

?

?

Figure 1.16: Illusions of belt-driven transmission.

for shafts and bearing and the change of objective characteristics (the extensitivity) of a belt?
The attempt to hide from explaining the length contraction mechanisms behind the common phrase of type: ”this is a kinematic eﬀect of space itself” is unsuccessful because of uncertainty of the ”contraction direction” (toward which point of space?). Really, the point of reference (the observer) can be placed at any point of the inﬁnite space – inside, to the left or to the right side from an object; and then the object as a whole will not only contract, but also move toward the given arbitrary point. This fact immediately proves the inconsistency or unreality of the given eﬀect. It is not clear, toward which end the segment will contract, if the moving system with two (moving) observers at segment’s ends was made impulsively. The situation can not also be saved by the phrase about the ”mutual uniqueness of Lorentz’s transformations”. This condition is quite insuﬃcient. The mutual uniqueness of some mathematical transformation allows one to use it for convenience of calculations, but this does not imply in any way, that any mutually unique mathematical transformation has physical sense. Also strange is the process of stopping of contracted bodies. The questions arise: toward what side do their dimensions restore? Where has the contraction

46
0011001100110011001100110011

CHAPTER 1. SRT KINEMATICS

Figure 1.17: Slipping inside the sandwich.

of space gone, if various remote observers could observe this body?
Problems on thin rods
Let us consider in detail the problem on 1-meter-long thin rod slipping over a thin plane having a 1-meter-wide hole [106] (see [33], exercise 54). It is rather strange, that any object should contract, turn or ”deﬂect and slip down” in exactly the same manner, as it is required for SRT to be ”saved” from contradictions at any cost (however, such an approach is an indirect recognition of principal indetectability of kinematic eﬀects of SRT). What relation to the given problem can have a real rigidity of a rod? None! Let the rod be slipping between two planes (a sandwich), so that only a part of a rod freely hanging over a hole be participating in deﬂection (Fig. 1.17). If the 1-meter rod can ”deﬂect and slip down” into the hole shortened down to 10 cm (or 10 times), then in exactly the same manner the 1-kilometer-long rod (which should not fall-through neither in the classical physics, nor even in SRT in the plane’s frame of reference) could also ”deﬂect and slip down” into the hole. The declarative mentioning of the velocity of acoustic oscillations (for the balance establishment mechanism) is the ”plausible” hiding of the truth. Let there are two identical real horizontal rods at the same height (Fig. 1.18). The ﬁrst rod slips over the desktop (at the pressed position) and begins to hang downwards with one tip at instant t = 0. At this instant (t = 0) the second rod begins to fall freely downwards. Obviously, for any time instant t > 0 the second rod will be displaced downwards (or fall) to a much greater distance as compared to the deﬂection of ﬁrst rod’s tip (and, actually, SRT tries to replace the real body by a body with zero rigidity). For analyzed problems the relativistic velocities can only decrease the rigidity eﬀect as compared to the case of low velocities, thus ever more approaching a real body to the model of absolutely solid

1.5 PARADOXES OF LENGTHS SHORTENING

47

010011010011000001111101001101001101001101001101001101001101001101001101001101001100000111110100110000011111010011

Figure 1.18: Rigidity and the deﬂect of a rod.

body. Indeed, the rod is deﬂected in the direction perpendicular to the relativistic motion. Therefore, this problem is similar to the problem on massive body slipping over thin ice on a river: at low velocities the body can fall through (breaching of ice due to its deﬂection), and at rather high velocities the body can slip over ice without falling through (the ice deﬂection is small). The rate of acoustic oscillations is much lower, than the speed of light. Therefore, the molecules manage to eﬃciently participate in rod’s deﬂection for shorter time as compared to the static case; that is, the deﬂection will be smaller. Let us take the width of the lower plane to be one molecule larger, than the displacement of rod’s deﬂection (for some particular preselected material). At the second end of a hole we shall make a very shallow taper of the plane (Fig. 1.17), so that the given rod could continue slipping over the plane (smoothly). Obviously if the rod does not slip down into the real 10-cm hole at nonrelativistic speeds, the more so the rod could not slip down into the hole allegedly shortened down to 10 cm at relativistic speeds. What will happen to the 20-cm or 1-km rod for all former characteristics of the plane? And if we, for the former geometrical characteristics of the experiment, will take various materials for a rod (from zero to maximum rigidity)? Obviously, with precise adjustment of all parameters for one case it is impossible to eliminate the contradiction for all remaining cases. For ”saving” SRT it is necessary either to postulate, that the rigidity in the experiment ceases to be an objective property of materials (but ad

48

CHAPTER 1. SRT KINEMATICS

Z
V X
Y
Figure 1.19: ”Turning” the rod.
hoc depends on the observer, geometric size and velocity), or to postulate, that the second end of a hole jumps up ad hoc in the ”necessary manner”. Does the goal justify similar means?
A similar problem on passage of a rod, ﬂying along axis X (now the rod is no longer pressed against the plane) through the niche of the same size (slowly running over the rod along axis Z) has even entered the popular literature [6]. The relativists ”eliminate” the contradiction in evidences of the observers by turning the rod in space (then the rod will pass through the niche in any case, as in the classical physics). However, the turning does not eliminate the Lorentzian contraction. Let us illuminate the niche from below along axis Z by the parallel beam of rays (for example, from a remote source). Let now rapidly pass the photographic ﬁlm high above the niche parallel to the plate, but perpendicular to the mutual motion of a rod and a plane, that is, along axis Y (Fig. 1.19). Then, in spite of rod passage, the result in SRT will all the same will be diﬀerent for diﬀerent observers. In the classical physics we would obtain the full darkening of the photographic ﬁlm at

1.5 PARADOXES OF LENGTHS SHORTENING

49

v

v

A

O

B

Figure 1.20: The paradox of two pedestrians.

the time of rod passage through the niche (this would be marked by a completely dark section on a light strip). A similar full darkening would take place in SRT from the viewpoint of the observer situated on a rod (since the niche will contract and turn). However, from the viewpoint of the observer situated on a plate (and on the photographic ﬁlm) the rod will contract and turn. Therefore, the full darkening will never take place. In such a case, who is right? There is the more dramatic situation with an angle of turning of the rod, since it depends on the relation of velocities. Let other small rod slide on our rod at some arbitrary velocity. Observers at the both rods will claim that the clearance between the rods is absent. However, according to the SRT, these rods must be turned at diﬀerent angles for an observer at the plate. There appears the evident logical contradiction.
Some remarks on lengths shortening
We shall additionally consider now the relativistic eﬀect of contraction of distances (the paradox of pedestrians). We shall ”agree in advance” about the following mental experiment (Fig. 1.20). Let a beacon, disposed at the middle of a segment, to send a signal toward its ends. Let segment’s length be one million light years. At the time of arrival of a ﬂash two pedestrians at segment’s ends begin to walk at equal velocity toward the same preselected side, along the straight line containing the given segment, and they will be walking for several seconds. The moving segment (a system of two pedestrians) should be contracted relative to the ends of a motionless segment by some hundreds kilometers. However, none of pedestrians will ”ﬂy away” for hundreds kilometers during these seconds. The moving segment could not also be torn oﬀ at the

50

CHAPTER 1. SRT KINEMATICS

middle, because the Lorentz transformation laws are continuous. So, where has this segment been contracted in such a case? And how this can be detected?
For ”justifying” the relativistic contraction of lengths Fock [37] discusses as follows. In the motionless coordinate system the lengths (factually ﬁxed by tips of a segment) can be measured non-simultaneously, but in the moving system they must be measured simultaneously. From the invariance of the interval
(xa − xb)2 − c2(ta − tb)2 = (x′a − x′b)2 − c2(t′a − t′b)2
at the choice of t′a = t′b, ta = tb we obtain |xa − xb| > |x′a − x′b|. But in such a case, why we can not choose ta = tb arbitrarily in order to obtain the objective length |xa − xb| in a unique manner? The existence of the process of measuring the length (the tips of a segment), which is independent of time and of the concept of simultaneity for the intrinsic frame of reference, proves a full independence of time and spatial characteristics in this system. But why for the other, moving system must arise any new additional link between the coordinates and time except the kinematic concept of velocity?
Wrong is Mandelshtam’s [19] judgement, that there is no ”real length”, and his example with the angular measure of an object. The angular measure of an object depends not only on object’s size, but also on the distance to it, that is, on two parameters. Therefore, this measure can be made unique only if one parameter – the distance to an object – is ﬁxed. Incorrect is also Mandelshtam’s statement, that in any method of measuring the lengths the rods moving in diﬀerent manner have different lengths. For example, possible is the procedure of measurement (direct comparison) of the rods previously turned perpendicular to the relative motion of the rods. Then the rods can be turned in arbitrary manner. They could even be slowly rotating in order to occur to be perpendicular to the motion at the time of coincidence. In such a case this method is completely independent on the relative motion even in SRT.
Some relativists believe that there is no length contraction at all – only the turning exists, for example, for a cube (i.e. they cannot unambiguously agree even between each other). The absence of real turning

1.6. THE RELATIVISTIC LAW FOR VELOCITY ADDITION 51
of a cube (or the fact that this eﬀect is only apparent) can easily be proved, if the cube will ﬂy being pressed against a ceiling. Generally speaking, the distance to objects, their visible velocity and size can be determined, even with the help of the light, by several techniques which are ”self-consistent” by themselves. For example, even for a single observer: from the angular size, from illumination, from the Doppler eﬀect. But the obtaining of diﬀerent values for the same physical quantity does not cancel at all the only true objective characteristics of a body and its motion (under which the instruments are calibrated).
The SRT tries to ”purchase” the consistency of its determination of lengths by refusal from the objectivity of some other physical quantities. However, this trick won’t ”work” with respect to the time – it is irreversible. Note some strange thing: in the sense of reversibility (in transition from one inertial frame of reference to the other and back!) the linear Lorentz transformation are fully equivalent both for coordinates and for the time (they are reversible). It seems strange, then, that a diﬀerence between bodies’ lengths vanishes with return at initial place (for twins, for example), but the disparity remains in the time elapsed.
1.6 The relativistic law for velocity addition
Recall that the kinematics does not study the causes of motion, but, for example, knowing the given velocities it ﬁnds the result of addition of these velocities. The issues of dynamics of particles (i.e. causes of motions) require independent consideration (see Chapter 4).
We begin with a remark concerning the relativistic law for velocity addition. For two systems participating in relative motion, the determination of their relative velocity causes no doubts (neither in classical physics nor in SRT). Let system S2 be moving relative to system S1 at speed v12; similarly, let system S3 be moving relative to S1 in the same directions at speed v13. In fact, the relativistic law for velocity addition deﬁnes the relative speed of that motion in which the observer does not participate himself: The speed of motion of system S3 relative to S2 is

52

CHAPTER 1. SRT KINEMATICS

determined as

v23

=

v13 − v12

1

−

v13 v12 c2

.

(1.5)

It is precisely this form (although usually v13 is expressed in terms of v12

and v23), which discloses the real essence of this law: it tells what relative

speed of systems S3 and S2 will be recorded by the observer in S1, if the

Einstein light-signal method is used for time synchronization and for

measuring length. Actually, we have here the ”law of visibility”. (For

the case of possible frequency dependence of light speed, this expression

will change – see Appendixes).

Consider the following methodological remark. One rather strange

kinematic notion from SRT is the non-commutativity of the rela-

tivistic law for velocity addition of non-collinear vectors. The non-

commutativity property (and the fact, that the Lorentz transformations

without rotations do not compose a group) is mentioned only brieﬂy

in some theoretical physics textbooks. By contrast, a similar property

in quantum mechanics essentially changes the entire mathematical for-

malism and physically expresses a simultaneous immeasurability of non-

commutating quantities.

It is seen from the general relativistic law of addition of velocities

that

v3

=

(v1v2)v1/v12

+ v1

+ 1 − v12/c2(v2 1 + (v1v2)/c2

− (v1v2)v1/v12) .

(1.6)

Clearly, the result depends on the order of transformation. For example,

in the case of sequence

+v1i, −v1i, +v2j, −v2j,
where i, j are the unit vectors of the Cartesian coordinate system, we obtain a zero sum velocity, and for the other order of the same quantities

+v1i, +v2j, −v1i, −v2j
we obtain a non-zero sum velocity, which depends on v1 and v2 in a rather complicated manner. The successive application of transformations (of motions) of v1i and v2j results in

v3 = v1i + 1 − v12/c2v2j,

1.6 ADDITION OF VELOCITIES

53

v1

v3

v clas

v’
3

v
2
Figure 1.21: Velocity parallelograms in SRT.
and in the other order of v2j and v1i it results in
v3′ = v2j + 1 − v22/c2v1i;
that is, we obtain diﬀerent vectors (Fig. 1.21). In such a case, what can the decomposition of the velocity vector into
components mean? First, the transfer of simplest, classical calculation techniques (the commutative algebra) to relativistic (non-commutative) equations is illegal: even the solution of vector equations in a componentby-component manner requires additional postulates, complications or explanations. Second, a simple application of the methods of classical physics (such as the principle of virtual motions, the variation methods, etc.) is impossible. In this case, even a ”zero” had to be ”individualized”: the number of ”zero” quantities, composed of some vector combination, should be equal to the number of ”zero” quantities composed of a mirror vector combination. Hence, the theory of ﬂuctuations would also require additional substantiation in such a case. Thus, contrary to the statement ”on the simplicity and elegance of SRT”, the correct justiﬁcation of even simplest procedures would require introducing many artiﬁcial complications and explanations (which are absent in

54
v

CHAPTER 1. SRT KINEMATICS
v
V

Figure 1.22: The law of addition of velocities and the paradox of a balance.

the textbooks).
Consider the logical contradiction of the relativistic law of velocity addition for the example of one-dimensional case. Let us have a balance in the form of a horizontal groove with a horizontal transverse pivot at the middle. Two identical balls of mass m will roll along a groove from the pivot to diﬀerent sides (Fig. 1.22).
To avoid discussing properties of the relativistic mass, we shall proceed as follows. Let the balance pivot be frictionless except when the balance is in the horizontal position (the ”dead point”). At this position, the threshold of the friction force does not allow the balance to rotate due to any possible small diﬀerence between the relativistic masses of the balls. But this sensitivity threshold cannot prevent the balance from rotating oﬀ the ”dead point” in the absence of one of balls – it will fall downwards. Let the velocities of balls in the system be equal in magnitude. Then the balls in this system will simultaneously reach the edges of the groove and fall downwards, so that the balance will be kept at the horizontal position. Consider now the same motion in the system, relative to which the balance are moving at speed V . Let be V → c only, but v ≪ vs, where vs is the speed of sound for the groove material. Then the balance can be considered as absolutely rigid (we can ignore acoustic waves). According to the relativistic law of addition of velocities,

v1

=

V −v 1 − vV /c2

,

v2

=

V +v 1 + vV /c2

.

1.6 ADDITION OF VELOCITIES

55

The motion of a middle point at speed

v1

+ v2 2

=

V

1 − v2/c2 1 − v2V 2/c4

<

V

always lags the motion of the balance. Thus, the ball moving against

the direction of motion of the balance will fall down ﬁrst. As a result,

the equilibrium will be violated, and the balance will begin to rotate.

So, we have a contradiction with ﬁrst observer’s data. Will the observer

be hit if he will stands under the right-hand part of the balance?

Will the Lorentz transformation laws be able to describe successive

transitions from one inertial system to another, and does the relativistic

law of addition of velocities correspond to real velocity variations? Cer-

tainly not. First, recall the meaning of the relativistic law of velocity

addition. It must prove that the addition of any motions cannot lead to a

speed greater than light speed. What is the manner (sense) in which mo-

tions are added in this case? For example, the Earth moves relative stars

(factually, there exists the ﬁrst reference system), a spacecraft ﬂies up

from the Earth with large velocity (in fact, the second reference system

is ”created”), then, another spacecraft ﬂies up from the ﬁrst spacecraft

(factually, the third reference system is ”created”), and so on. It is just

the meaning for consecutive transformations. Then the following ques-

tion no longer arises: in the relativistic law of velocity addition, which

velocity must be considered as the ﬁrst one, and which velocity is the

second one (This is important for non-commutative transformations).

All the examples in this Section have this meaning.

Let us consider now the Lorentz transformation law for arbitrary

directions of motion:

r1

=

r

+

1 V2

1

− 1 (rV)V +

Vt ,

1 − V 2/c2

1 − V 2/c2

t1 =

t + (rV)/c2 . 1 − V 2/c2

It can easily be veriﬁed, that the successive application of the relativistic law of velocity addition (1.6) to quantities

v1i, v2j, −v1i − v2 1 − v12/c2j

(1.7)

56

CHAPTER 1. SRT KINEMATICS

will give a zero. To an arbitrary vector r = xi + yj we apply the Lorentz

transformation laws successively with the same set of velocities. Then

we have:

r1 =

x 1

+ −

v1t v12/c2

i

+

yj,

t1 =

t

+ xv1/c2 1 − v12/c2

.

Further, we have:

r2 =

x 1

+ −

v1t v12/c2

i

+

y

1

− 1

v12/c2 + − v12/c2

v2t 1

+ −

xv1v2 v22/c2

/c2

j,

t2

=

t

+

xv1/c2 + yv2 1 − v12/c2

1 − v12/c2/c2 1 − v22/c2

.

We shall not write down the expressions for r3 and t3 in the explicit form because of their awkwardness. However, using graphical programs, we can be convinced of the following properties: 1) In the new system, the initial time is desynchronized at any point of space except the coordinate origin. 2) The time intervals have changed: dt3 = dt; that is, we got into a new moving system, rather than into the initial resting one. Therefore, in the textbooks, as a minimum, the meaning of the Lorentz transformation laws or of the relativistic law of velocity addition is uncovered rather incorrectly. 3) Line segments occur to be not only changed in length, but also turned around. We can easily be convinced of this, if we ﬁnd numerically the angle of rotation; i.e. the diﬀerence

α = arctan

y3[x(1), y(1), t] − y3[x(0), y(0), t] x3[x(1), y(1), t] − x3[x(0), y(0), t]

−arctan

y(1) − y(0) x(1) − x(0)

.

These properties can be discussed mathematically in terms of the ”pseudo-Euclidean character of the metric” as much as you like. However, physically the situation is quite simple. These properties prove

1.6 ADDITION OF VELOCITIES

57

the non-objective (i.e. only illusory) character of the Lorentz transformation laws and of the relativistic law of velocity addition, and their disagreement with each other. Indeed, since we have successively passed from one inertial system to another, and the rotation implies the noninertial character of a system, SRT itself escapes the limits of its own applicability; i.e., it is inconsistent. If this rotation were real, this would imply a non-objective character of the inertial system notion (since the result would depend on the method of transition to the given system) and, as a consequence, the lack of a proper basis for SRT to exist.
Let us try to clear up why it is that treatments from the textbooks result in disagreement between two expressions, the relativistic law of velocity addition and the Lorentz transformation laws, in spite of the fact that the ﬁrst expression is derived from second one. Recall the following derivation for the example of one-dimensional mutual motion of systems K and K′. Proceeding from the Lorentz transformation laws

x1 =

x+Vt , 1 − V 2/c2

t1 =

t + V x/c2 1 − V 2/c2

we divide the diﬀerential dx1 by dt1 with regard to deﬁnitions v = dx/dt and v1 = dx1/dt1 and obtain:

v1

=

1

v +

+V vV /c2

.

This indicates the following things: 1) The observer is at the origin of system K and measures the distance x to the studied body in its system K. 2) He considers time t to be universal in his system and determines the velocity of a body in his system v = dx/dt. 3) He measures speed −V of system K′ with respect to K using his own (!) time t, and considers the relative velocities of systems to be mutually opposite in direction. This observer cannot measure any other thing: the summary velocity v1 is a computable quantity. Thus, we came to the treatment [49] given above: the relativistic law of velocity addition determines the velocity of that relative motion, in which the observer does not participate by himself. This eﬀect is not real, but only

58

CHAPTER 1. SRT KINEMATICS

apparent (when we use some particular rules of SRT). In the essence of the formula, we cannot simply pass to the second substitution for determining v2, though, formally, any arbitrary number of velocity values can be sequentially substituted into the expression for the relativistic law. In the case of addition of motions along a single straight line, the classical property of commutativity conserves, and the contradiction is veiled over. But if the velocity vectors are non-collinear, then item 3) becomes untrue, and the inconsistency and disagreement of the law of velocity addition and Lorentz transformation laws are immediately exhibited.
But we can apply another approach to the example discussed previously: we shall search for the sequence of three transformations of velocities that retains the initial time in the Lorentz transformation laws invariant. Then it can easily be veriﬁed that, instead of (1.7), a single succession can be taken:

v1i, v2j, −v1 1 − v22/c2i − v2j.

(1.8)

However, at ﬁrst, the turning of segments remains. Second, a new set of velocities does not satisfy, in the given succession, the law of velocity addition, i.e. factually there changes the order of substitution of the velocities v1 and v2 in the law of velocity addition (that is inconsistent with the essence of this law). Therefore, the contradictions are not eliminated in this case as well. The Thomas precession is an example of SRT inconsistency also: starting from the sequence of inertial systems (moving rectilinearly and uniformly), the resulting rotation of objects is suddenly obtained (principally noninertial motion). Thus, the passage from the Lorentz transformations (outlined in standard textbooks) of ”mathematical space” 1 + 1 (t + x) to the Lorentz transformations of 1 + 2 ”space” (or 1 + 3) leads to physical contradictions.
Many intuitively clear properties of physical quantities lose their sense in SRT. For example, the relative velocity ceases to be invariant. The particles, ﬂying away along the same straight line at various velocities, form in SRT a complicated ”fan of velocities” for a moving system. The isotropic velocity distribution in SRT ceases to be the same for the other moving system. No declared simpliﬁcation does exist in SRT in reality.

1.6 ADDITION OF VELOCITIES

59

The impossibility of existence of velocities v > c in no way follows from SRT. And the addition, that this statement relates to the signal transmission rate only, is only artiﬁcial addition (because of existence of obvious counterexamples to the extended treatment). However, the notion of signal (information) remains insuﬃciently determinate even with a similar addition. For example, while receiving a signal from the ﬂare of supernova, are we not sure that the same information ”is contained” at the diametrically opposite distance from the supernova (that is, we know about it at velocity of 2c)? Or this is not information? Therefore, SRT can only deal with the information on a material carrier of electromagnetic nature propagating in vacuum sequentially through all points of space from the signal source to a receiver.
Let us make some comment on ”astonishingness” of the relativistic law of ”addition” of velocities, which allows to exchange light signals even for the algebraic sum of velocities greater than c. We pay attention to the obvious fact: for exchanging information the signals should be sent necessarily in the direction of an object, rather than in the opposite direction. Therefore, there is nothing surprising in exchanging the signals, where in the classical case it occurs also that, as a result of formal addition of velocities, v1 + v2 > vsignal. Let two airplanes to take oﬀ from the aerodrome O at velocities of 0.9vsound and ﬂy away from each other in the opposite directions of axis X (the relative velocity is 1.8vsound). Whether the exchanging of signals between them is possible? Certainly yes! Because the sound wave propagates in air irrespective of the velocity of source S1 at signal issuing time, the ﬁrst airplane (which has sent a signal) will catch up the wave front propagating in the positive direction of axis X, whereas the second airplane will ”compete” with the wave front propagating in the negative direction of axis X. Both airplanes are moving slower as compared to propagation of corresponding wave front sections nearest to them (see Fig. 1.23). Thus, the sum of velocities is compared (in a complicated manner), in reality, with quantity 2vsound, rather than with the speed of sound (and for light – with the value of 2c).
It is obvious also that physical restrictions on the value of speeds cannot be applied by mathematics (by the fact that in some expressions

60

CHAPTER 1. SRT KINEMATICS

2

S1

1

O

X

Figure 1.23: Exchange of a signal.

there exists a negative value under the radical sign). It should be remembered that all SRT expressions are introduced with use of a light signal exchange (the method of Einstein’s synchronization). But if a body moves faster than light long since, it simply cannot be caught up by signal sent in pursuit. In a similar manner, a synchronization can be made with use of sound (expressions with radicals could be written), but the impossibility of supersonic speeds in no way follows from here at all.
1.7 Additional criticism of relativistic kinematics
We shall begin with some general remarks. The group properties of mathematical equations, as the transformations with mathematical symbols, do not bear any relation to any physical principles or postulates; that is, the group properties can be found without additional physical hypotheses. For example, the Lorentz transformation laws, which reﬂect the group properties of the Maxwell equations in vacuum (or of the classical wave equation, including that in the acoustics), are not bound at all with SRT’s postulate of constancy of the speed of light or with the relativity principle.
The theory of relativity is, in fact, ”the theory of visibility”: it is about what we see in an experiment, if it is based (with generalization for space and time properties) on the laws of electromagnetic interactions (the absolutisation of electromagnetic phenomena). Similarly, the

1.7 ADDITIONAL CRITICISM

61

S

S’

v

O

A

A1

O’

A’

Figure 1.24: The non-locality paradox.

question can be raised: What will the phenomena observed by means of sound, etc., look like? Certainly, the ﬁniteness of the rate of transmission of some interactions alters the phenomena observed with the help of these interactions. But this circumstance does not prevent making unique extrapolations for ”binding” to space and time (which are the absolute physical notions) for the unique description of the world without limitation by any ”overall” hypotheses.
Newtonian space possesses an important property: systems with lower dimensions can possess similar properties. For example, a vector can be introduced not only in three-space, but also on a plane and on a straight line. In RT, three-dimensional quantities do not possess vector properties (only the 4-vectors do this); that is, there is no continuous limiting transition to classical quantities (the ”nearly vector” → vector).
As the next remark, we shall describe the ”non-locality” paradox. Note that all SRT formulas do not depend on the previous history of motion, i.e. they are local. Let system S′ move at velocity v relative to system S. Let a light ﬂash occur at center O at the time of its coincidence with center O′. At time t in system S, let the wave front reach point A, and in system S′ – point A′, respectively (Fig. 1.24). Now we impart, by pulse, velocity v to a signal receiver in system S at point A1 = A′. It happened that the wave front has moved right away to A′

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CHAPTER 1. SRT KINEMATICS

(since we are now in system S′). But where had the wave front been at the same time instant? Did the time at A1 = A′ change? And if we will stop the receiver at A1 after a moment? The time will be restored, and the wave front will again return to A? And the observer will forget that he saw a ﬂash of light? Then, in order to see the future, one must move faster? The fact that the observer at A1 had not at all times moved together with system S′ explains nothing, since another observer, who had all the time moved together with system S′, could be at A′. Does it occur that one of them will see the event, whereas the other one will not? If so, the objective nature of science disappears.
The next additional issue is as follows. Does a wave packet (light) move in vacuum at light speed? If yes, then we cannot break it down into (separate) pulses (signals) by means of a stroboscope: due to length shortening, the length of each pulse and the length of each interval between the pulses must be zero (which is contradictory). If, however, we suppose the dimensions of obtained pulses (signals) and intervals between them to be ﬁnite in the resting (laboratory) coordinate system, then in the intrinsic reference system of package, both pulses and intervals should be inﬁnite (but how can we interrelate in this case the pulse and the interval, where it is absent?). In essence, it is the following question: whether light and the space between signals are material or not?
Let us make now some comment about a change of the visible direction of particle motion or about a change of the visible direction of wave signal arrival (remember the aberration, for example) as an observer goes to other moving system. This simple classical fact is described in SRT as the turn of all wave front at some angle. As this takes place, the wave front presents a light sphere at the same time instant. We would remind that the wave front in SRT is diﬀerent at the same time instant for systems moving relative each other (just as the result of a change in running of time). However, the prehistory of motion of recording instruments is included in none SRT formula. If a photon has been ﬂying in space between a source and a receiver, it is causally connected in no way with motion of the source and the receiver at the same time instant. The interaction of the recording instrument with the photon occurs just

1.7 ADDITIONAL CRITICISM

63

v
Figure 1.25: The change of the direction of perceiving motion.
at the time of signal reception only. No diﬀerence exists whether the receiver had been having a velocity v all the time and was brought into this space point at the time of signal reception, or it had been being motionless at the same space point, but acquired the same velocity v at the instant before signal reception (the result of interaction with the photon were the same in both cases). Thus, the only fact of photon arrival to the given place of space matters for the fact of receiving of a photon as such. Obviously, the value of some velocity at the given place of space does not change the fact of signal arrival as such (but, according to the Doppler eﬀect, its frequency will be changed only). If the fact itself of the signal receipt were dependent on this, then what does the substitution of values in the Doppler formula at the one of observation systems mean? Therefore, no real turn of all wave front can be (since it reﬂects the fact of signal arrival). This is the local (at the given point) mathematical (diﬀerential) method to determine the visible direction of signal reception. It can be easily understood by analogy with the usual natural phenomena – with rain and snow (Fig. 1.25). If you look at a cloud over your head in the windless weather when it is raining, you see vertical fall of drops (the direction of ”signal” reception). But if you will

64

CHAPTER 1. SRT KINEMATICS

run (it is better to remember a car travel in a snow day), the direction of drops arrival (the direction of ”signal” reception) will far ahead along motion and can be lack of coincidence with the real cloud. However, the horizontal front of rain either already reached the earth surface (the fact of ”signal” reception), or not, and this fact does not depend on your motion at the given point of the earth surface at all (see Fig. 1.25).
Let us discuss some speculative constructions of SRT. So, unreal in SRT is the consideration of inﬁnite systems, such as a conductor with current, in ”explaining” the appearance of additional volume charge (the game with inﬁnities). In reality, the conductor can be close-loop (ﬁnite) only. In this case the explanation is not only complicated methodically, but also contradictory. Let us consider a square loop with current (for example, a superconducting loop). The value of a charge of each electron and ion is invariant; the total number of particles is invariable too. How can change the density of charges in this case? Consider the motion of electrons from the viewpoint of a ”system of ionic grid” (Fig. 1.26). According to SRT, the ”electronic loop” should decrease in size (the contraction of lengths because of motion of electrons on each rectilinear section). It would seem that, owing to symmetry of the problem, the ”electronic loop” should enter inside the ”ionic loop”. However, in such a case we would have a strange asymmetrical ﬁeld (of dipole type) near the conductor. Besides, while moving at high velocity, the electrons and ions could appear on diﬀerent sides from the observer. It is completely unclear, how such a transition through the observer (perpendicular to the motion of particles!) could take place at all? And by what forces the charged electrons (as well as the ions) would be retained together in a ﬂux, not ﬂying away to diﬀerent sides? Even if we take advantage of the ﬁtting SRT uncertainty (towards what end does the contraction occur?) for one side of a square, all questions still remain for its other sides.
The SRT’s system of watches and rules is inconvenient both theoretically and practically, since it supposes that all the data are gathered and analyzed (interpreted!) somewhat later. The uniqueness of interrelation between the classical Newtonian and relativistic Lorentzian coordinates does not imply automatic consistency of latter ones (just in this, phys-

1.7 ADDITIONAL CRITICISM

65

?

?

?

?

j

?

?

?

?

?

?

?

j

?
j
?
?

Figure 1.26: The paradox of loop with current.

ical sense consists the distinction of physics from mathematics). For example, we could use in all SRT formulas the speed of sound in air instead of speed of light and consider the motions on the Earth at subsonic velocities in resting air. However, the inconsistency of similar transformations (for the time) would be immediately revealed in the experiment. This fact demonstrates the hazard of formally mathematical analogies for physics.
It is obvious that the relativistic hypothesis for time dilation is wrong, since only the square of relative velocity is included in the formula (the eﬀect does not depend on the velocity direction). Take 4 identical objects. Let second object be moving at some velocity v12 relative to the ﬁrst one, then its time will be slowed down relative to the time of the ﬁrst object. You say that it is an objective eﬀect? (We would remind the meaning of the word ”objective”: an eﬀect does not depend on presence and properties of the observer which not interacts with the object under study.) We even would not ﬂy to check it. Let

66

CHAPTER 1. SRT KINEMATICS

the third object be moving relative to the second one in an arbitrary direction with some velocity v23. Then, by analogy, its time will be slowed down relative to the time of the second object. Is this eﬀect objective again? Let the fourth object be motionlessly placed near the ﬁrst object. We even does not try to debate with which velocity the fourth object moves relative to the third object: it is important only that in the general case this velocity does not zero. Therefore, again we have ”objective relativistic” time dilatation of the fourth object relative to the third one. Thus, dt1 > dt2 > dt3 > dt4. But dt1 = dt4 (and we have no need to ﬂy somewhere), since the fourth and ﬁrst objects are in relative rest! Similar absurdity was obtained as the result of a fanatic relativistic faith in the uniqueness and infallibility of Einstein’s method of synchronization in pairs. Objectiveness melt away from under feet, and a remainder is either the relativistic seeming eﬀect or pure rated combinations (”ﬂoating time belts”). What matter is the declared greatness?
Now we shall make some general remarks. The whole SRT kinematics follows from the invariance of the interval dr2 − c2dt2 = inv. However, we see that this expression is written for the empty space. In a medium the speed of light is non-constant, it can be anisotropic, and the light of non-arbitrary frequency can propagate in the given particular medium (remind the attenuation, absorption, reﬂection, dissipation). There is no sections of physics, where the properties of phenomena in vacuum would be automatically transferred to the phenomena in other media (for example, in liquids – hydrodynamic and other properties; in solid bodies – elastic, electrical and other properties). That is, they are not determined by the properties of the empty space. And only SRT pretends to a similar universal ”cloning” of properties.
Generally speaking, the properties of light, which are intrinsically contradictory and mutually exclusive, are simply postulated in SRT. Therefore, wrong is Fock’s [37] statement, that the light is a simpler phenomenon, than the rule. It is not worth to extol the role of light signals and all ”visible things”; otherwise a teaspoon inside a glass with water could be considered as the broken one (pure geometrically, the fallacy in this consideration can easily be tested by the direct location

1.7 ADDITIONAL CRITICISM

67

of coordinates of all ”teaspoon outlets” at the boundary of the liquid). The classical time (or the time determined by an inﬁnitely remote source at the middle perpendicular to the line of motion) possesses some important advantage: we know a priory that it is identical everywhere, and no calculations or discussions are required concerning the prehistory of the process or properties of the space. Actually, SRT uses the speed of light as one of measurement standards. Remind that in the classical kinematics there are two measurement standards: the length and time (we will ”formulate” evident ”laws of constancy of standards”: the length of the standard of 1 m is constant and is equal to one meter, the duration of the standard of 1 sec is constant and is equal to one second, but relativists din ”the Great Law of Constancy of the relativistic standard” into everyone’s ears). Since the introduction of a standard is the deﬁnition, its properties are not subject to discussion [19]. As a result, everything, which is related with the light propagation, ceases to be a prerogative of experiment in SRT. And because all derivations in SRT are written only for the events – the light ﬂashes, then SRT occurs to be logically inconsistent (to say nothing of the fact, that the ”use” of properties of light in vacuum is profusely spread to all other ”non-vacuum” phenomena).
Feynman in his book [35] says with sarcasm about the philosophers and about the dependence of results on the frame of reference, but he does not emphasize that, in spite of any ”appareness”, the subjects have real objective characteristics. For example, a man may seem to have a size of ant from the great distance, but this does not mean that he has really reduced (all instruments are used to be calibrated just under objective characteristics). The reasoning on a relativity of all quantities seems to be realistic, but (!) once the time in SRT became relative and the rate of interaction was supposed to be ﬁnite, the notion of relative quantity for spatially separated objects has become indeﬁnite (It depends on the path of connection, is not bound causally, depends on the system of observation, etc). The deﬁnition of all quantities with respect to ”far stars” is senseless, since we can see a ”never existing reality”. For example, the Alpha-Centaur has been at this particular place and possessed such properties 4 years ago; the other stars have been the same

68

CHAPTER 1. SRT KINEMATICS

as we see tens or hundreds years ago and the distant Galaxies – billions years ago, i.e. the signal was sent when the earth observer did not exist yet, and is accepted when, possibly, the source itself no longer exists. In such a case, relative to which should we determine the quantities? It is clear that the relative quantities can be determined only with respect to the local characteristics of space (the unique instantaneous causal bond).
Some important remark concerns the notion of relativity, which has even entered the name of the SRT theory. Contrary to Galileo’s ideas on isolated systems, an interchange of light signals between systems is used in SRT. The notion of relativity has been worked up to nonsense in SRT and lost its physical sense: in fact, the system with several (as a rule, two) objects is singled out, and the whole remaining real Universe is eliminated. If such an abstraction can even be postulated in SRT, then, the more so as, one can simply postulate the independence of processes inside the separated system on the velocity of system motion relative to the ”emptiness” which remained from the whole Universe. But, even in spite of such an abstraction, no ”real” relative quantities will appear for bodies (such as rij, vij, etc.). Indeed, the response of body i to the attempt of changing its state is determined by the local characteristics: the state of a body i and the state of the ﬁelds at the given point of space. But the changes having occurred with body i will have an eﬀect on the other bodies j only in some time intervals ∆tj. Thus, all changes of quantities should be determined relative to the local place (or local characteristics). And these phenomena just represent manifestations of the Newtonian absolute space. The question, whether the separated direction and separated coordinate origin (either moving or resting) exist in this absolute space – is quite diﬀerent question. In the abstract (model) theories this question can be postulated, for example, from the considerations of convenience of the theory; but for our unique real Universe it should be solved experimentally. The absolute time notion in the classical Newtonian physics was extremely clear as well. The time should be uniform and independent of any phenomena observed in a system. Exactly such a property is inherent in the time synchronized by an inﬁnitely remote periodic source on a middle perpendicular. However,

1.7 ADDITIONAL CRITICISM

69

in SRT the time is not an independent quantity: it is associated with the state of motion of a system v and with the coordinates, for example, by the relation c2t2 − r2 = constant. For uniform running of time the choice of the time reference point is arbitrary. For uniﬁed description of the phenomena and for comparability of the results the scales (units of measurement) should be identical for all systems. The time running uniformity automatically ensures the greatest simplicity of description of the phenomena and for the basis notion of time allows to introduce its standard deﬁnition.
Let us make some more methodical comments. Generally speaking, in SRT the method of comparison of the phenomena in two various inertial systems supposes, that both these systems have existed for inﬁnitely long time. However, the systems have often been ”linked” to particular bodies and have existed for a ﬁnite time only. Then, in each particular case the question needs to be studied: whether the prehistory of formation of these systems (its inﬂuence) has been ”erased” or not?
The Euclidean analogies with projections in the book [33] are completely inadequate to the reality. The projection is only an abstract method of description, the subject itself does not change at turning. In SRT, on the contrary, the characteristics of an (even remote) object instantaneously change with changing the motion of an observer (!).
The limiting transition from the Lorentz transformations to the Galileo transformations (for the time t = t′ + vx′/c2) indicates that the Newtonian mechanics is not simply a limit of low velocities β = v/c ≪ 1, but the other condition is required, namely: c → ∞. But in this case for many quantities in SRT there is no limiting transitions to classical quantities (see below, or [50]). However, in the classical physics c = ∞: its ﬁnite value was measured even in 17th century!
The property of maximum homogeneity of the space-time can be an attribute of either ideal Newtonian mathematical space and time (being actually a ”superstructure from above”), or of the model space (for example, with remotely non-interacting material points). The attempt to rest upon the mentioned property in RT as on the principal property of the real space and time is artiﬁcial. First, even in the earth scales we can not arbitrarily change the points of space, time instants, directions

70

CHAPTER 1. SRT KINEMATICS

of axes and velocities of inertial systems (recall the limited nature of the Earth space, the rotation of the Earth, the gravitational ﬁeld, the eﬀect of the Moon, the electric, magnetic, temperature ﬁelds and so on). We have listed above the real achieved practical limitations, rather than the principal restrictions somewhere at relativistic velocities and huge scales of the Universe. True, in the scales of the Universe with its real objects and gravitational ﬁelds this property is not conﬁrmed too: the model of uniform ”jelly” does not describe the real Universe. Second, in addition to the form of equations, the solution is still determined mathematically by the boundary and initial conditions. This also actually, on real ﬁnite scales, prevents any shifts and changes (or it is necessary to change, in addition, the imposed conditions). How can we approach the existing nonlinear properties and equations with the RT claims? Even the ”relativity” notion itself does not allow us to generalize (more likely, to narrow down) the real space with gravity. (As Fock [37] has emphasized, the ”general relativity theory” term is inadequate).
Theoretically, the principle of relativity (in any known form) supposes that ”without looking” outside the limits of a system it is impossible to discover its uniform motion. Earlier it was the ether, which has played a part of the all-penetrating medium for possible discovering such a motion. Note that the question was not about the discovery of the absolute motion, but only about the motion relative to ether. That is, it would be possible to compare these motions ”without looking” outside (here we keep in mind the calculating possibility only, since the system of registration points and standards cannot be tied with the ether). But even with ”canceling” the ether, according to the modern concepts, still remains the ”candidate” with similar properties – the gravitational ﬁeld (which is principally non-shielded). For example, from the relic radiation anisotropy, under the additional hypothesis on the equality of the rate of propagation of gravitational interactions and speed of light, may follow the anisotropy of the (all-penetrating) gravitational ﬁeld. Thus, the non-equal rights of inertial systems in macroscales can be found, in principle, ”without looking” outside even at the local point. This can be avoided theoretically under the hypothesis, that the rate of propagation of gravitational interactions is much higher than the speed of light; in

1.8. CONCLUSIONS TO CHAPTER 1

71

such a case the isotropy could be set up, but in actual practice – it is the prerogative of the Experiment.

1.8 Conclusions to Chapter 1
The given Chapter1 is basically devoted to general physical issues and to the systematic criticism of the relativistic kinematics. In so doing, a lot of logical and methodical contradictions of SRT is analyzed in detail. If only methodical inaccuracy were included in this theory, it could be corrected, some additional explanations, revisions, additions, etc., could be introduced. However, the presence of logical contradictions brings ”to nothing” any results of any theory, and SRT is not an exception in this respect (although rather undemanding attitude to SRT as compared with any other theory is evidenced in science).
We will brieﬂy summarize all of the preceding. In present Chapter such fundamental notions as ”space”, ”time” and ”relativity of simultaneity” were analyzed in detail. The logical inconsistency of the fundamental notion of ”space” in SRT was demonstrated on the basis of the following contradictions: the modiﬁed twins paradox, the paradox of n twins, the paradox of antipodes, the time paradox etc.. Then, the possibility of introducing a single absolute time independent of the velocity of motion was demonstrated by means of a periodic, inﬁnitely remote source situated across the plane (line) of motion.
Further, for numerous examples the inconsistency of the relativistic concept of length was demonstrated. (These examples include: the motion of a cross, rotation of a circle, lengths shortening, the belt-driven transmission, the indeﬁniteness of the direction of contraction, a loop with current, etc.). The SRT contradictions for the problems of rod slipping over a plane and of ﬂying rod turning, the non-locality paradox, limiting transition to classics, and so on, were considered in detail.
In Chapter 1 the true sense of the Lorentz transformations and of the interval invariance was discussed. The contradiction between the ”relativity of simultaneity” and the ﬁeld approach, founded upon the ﬁniteness of the rate of interactions, was considered in detail. The contradictions between the Lorentz transformations and the relativistic law

72

CHAPTER 1. SRT KINEMATICS

of velocity addition were also discussed in detail. Besides, in Chapter 1 the hyperbolization property of the ”relative quantity” concept itself and the space-time homogeneity properties were critically discussed in detail.
The ultimate conclusion of the Chapter consists in the necessity of returning to classical notions of space and time, to the linear law of velocity addition, and classical meaning for all derivative values. The questions of experimental veriﬁcation of SRT kinematics and questions concerning the relativistic dynamics will be considered in detail in Chapters 3 and 4 respectively. The questions of kinematics of noninertial systems will be touched in the next Chapter 2.

Chapter 2
The basis of the general relativity theory
2.1 Introduction
The logical inconsistency of kinematics of the special relativity theory (SRT) was proved in previous Chapter 1. This forces to return to the classical notions of space and time. Since relativists declare that SRT is the limiting case of the general relativity theory (GRT) in the absence of gravitation, then there arise some doubts in validity of GRT kinematics also. Unlike SRT, the GRT contains some rather interesting ideas, such as the principle of equivalence expressed via the idea of ”geometrization”. (Note that incorrectness of geometrization of electromagnetic ﬁelds is obvious: experiments show that neutral particles do not respond to the ”electromagnetic curvature of space”.) If it’s basis were true, the GRT could have a claim on status of a scientiﬁc hypothesis about some correction to the static Newton’s law of gravitation. Since it is not the case, the gravitation theory must be constructed in a diﬀerent manner. For the sake of justice it could be mentioned that GRT, in contrast to SRT, never were the universally recognized non-alternative theory. The current of true criticism of this theory has been continuing from its origin. There exist several rather advanced alternative theories (for example, [11,18] etc.). Although we shall not analyze theories
73

74

CHAPTER 2. GRT BASIS

other than GRT, it must be emphasized that theories, ”playing” around change of space and time properties and having relativistic kinematics of SRT as its limiting case, are obviously doubtful.
The basic purpose of present Chapter 2 is the criticism of basis notions of GRT. A logical inconsistency of space and time notions in GRT is demonstrated here. The (plausibly hidden) errors and disputable points from the textbooks [3,17,39] are displayed step by step in Chapter 2. In addition to conventional GRT interpretations, we shall also consider some ”relativistic alternative” to cover possible loop-holes for salvation of this theory. The time synchronization issues and the Mach principle are also discussed, and the attention is given to doubtful corollaries from GRT.

2.2 Criticism of the basis of the general relativity theory
Many GRT inconsistencies are well-known: 1) the principle of correspondence is violated (the limiting transition
to the case without gravitation cannot exist without introducing the artiﬁcial external conditions);
2) the conservation laws are absent; 3) the relativity of accelerations contradicts the experimental facts (rotating liquids under space conditions have the shape of ellipsoids, whereas non-rotating ones – the spherical shape); 4) the singular solutions exist. (Usually, any theory is considered to be inapplicable in similar cases, but GRT for saving its ”universal character” begins to construct fantastic pictures, such as black holes, Big Bang, etc.).
General remarks
Let us consider the general claims of the GRT. We begin with the myth ”on the necessity of the covariance”. The unambiguous solution of any diﬀerential equation is determined, except the form of the equation, also by speciﬁcation of the initial and/or boundary conditions. If they

2.2 CRITICISM OF GRT BASIS

75

are not speciﬁed, then, in the general case, the covariance either does not determine anything, or, at changing the character of the solution, can even result in a physical nonsense. If, however, the initial and/or boundary conditions are speciﬁed, then with substitution of the solutions we obtain the identities, which will remain to be identities in any case for any correct transformations. Besides, for any solution it is possible to invent the equations, which will be invariant with respect to some speciﬁed transformation, if we properly interchange the initial and/or boundary conditions.

The analogies with subspaces are often used in the GRT; for example, a rolled ﬂat sheet is considered. However, the subspace cannot be considered separately from the space as a whole. For example, in rolling a sheet into a cylinder the researcher usually transfers, for convenience, into the cylindrical coordinate system. However, this mathematical manipulation does not inﬂuence at all the real three-dimensional space and the real shortest distance.

The simplicity of postulates and their minimum quantity do not still guarantee the correctness of the solution: even the proof of equivalence of GRT solutions is a diﬃcult problem. The number of prerequisites should be, on one hand, suﬃcient for obtaining a correct unambiguous solution, and, on the other hand, it should provide wide possibilities for choosing mathematical methods of solution and comparison (the mathematics possesses its own laws). The GRT, along with artiﬁcial complication of mathematical procedures, has introduced, in fact, the additional number of ”hidden ﬁtting parameters” (from metrical tensor components). Since the real ﬁeld and metrics are unknown in GRT and are subject to determination, the result is simply ﬁtted to necessary one with using a small amount of really various experimental data (ﬁrst we peeped at the ”answer”, then we will believe with ”a clever air” that it must be in the theory in just the same manner).

Whereas in SRT though an attempt was made to conﬁrm the con-

stancy of light speed experimentally and to prove the equality of inter-

vals theoretically, in GRT even such attempts have not been undertaken.

Since in GRT the integral

b a

dl

is

not

meaningful

in

the

general

case,

since the result can depend on the path of integration, all integral quan-

76

CHAPTER 2. GRT BASIS

tities and integral-involving derivations can have no sense.

A lot of questions cause us to doubt as to validity of GRT. If the general covariance of equations is indispensable and unambiguous, then what could be the limiting transition to classical equations, which are not generally covariant? What is the sense of gravitation waves, if the notion of energy and its density is not deﬁned in GRT? Similarly (in the absence of the notion of energy), what is meant in this case by the group velocity of light and by the ﬁniteness of a signal transmission rate?

The extent of the generality of conservation laws does not depend on the method of their derivation (either by means of transformations from the physical laws or from symmetries of the theory). The obtaining of integral quantities and the use of integration over the surface can lead to diﬀerent results in the case of motion of the surface (for example, the result can depend on the order of limiting transitions). The absence in GRT of the laws of conservation of energy, momentum, angular momentum and center of masses, which have been conﬁrmed by numerous experiments and have ”worked” for centuries, cause serious doubts in GRT (following the principle of continuity and eligibility of the progress of science). The GRT, however, has not yet built up a reputation for itself in anything till now, except globalistic claims on the principally unveriﬁable, by experiments, theory of the evolution of the Universe and some rather doubtful ﬁttings under a scarce experimental base. The following fact causes even more doubt in GRT: for the same system (and only of ”insular” type) some similarity of the notion of energy can sometimes be introduced with using Killing’s vector. However, only linear coordinates should be used in this case, but not polar ones, for example. The auxiliary mathematical means cannot inﬂuence, of course, the essence of the same physical quantity. And, ﬁnally, the non-localizability of energy and the possibility of its ”spontaneous” nonconservation even in the Universe scales (this is a barefaced ”perpetuum mobile”) cause us to refuse from GRT completely and either to revise the conception ”from zero”, or to use some other developing approaches. Now we shall pass from general comments to more speciﬁc issues.

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77

The geometry of space
The question on the change of real space geometry in GRT is fully aberrant. The ﬁniteness of the rate of transmission of interactions can change only physical, but not mathematical laws. Whether shall we assert, that the straight line does not exist, only because its drawing into inﬁnity, even at light speed, will require inﬁnite time? (The same is true for the plane and space). The mathematical sense of derivatives can not change as well. One of GRT demonstrations ”on the inevitability of the change of geometry in the non-inertial system” is as follow: in the rotating coordinate system, due to contraction of lengths, the ratio of the length of a circle to its diameter will be lower, than π. Note that nobody can draw a ”new geometry” for this case: ”non-existing” cannot be pictured. In fact, however, not only the true, but even the observed geometry will not change: whether the mathematical line will move or change as we move? Although the radius, which is perpendicular to the circle motion, must be invariable, nevertheless, we suppose at ﬁrst, that the circle will move radially. Let we have three concentric circles of almost the same radius (Fig. 2.1). We place the observers on these circles and number them in the order from the center: 1, 2, 3. Let the second observer be motionless, whereas ﬁrst and third ones are rotating around center O clockwise and counter-clockwise at the same angular velocity. Then, owing to the diﬀerence in relative velocities and contraction of lengths, the observers will interchange their places. However, when they happen to be at the same point of space, they will see diﬀerent pictures. Indeed, the 1-st observer will see the following position from the center: 3, 2, 1, whereas the 2-nd observer will see the diﬀerent order: 1, 3, 2, and only the 3-rd observer will see the original picture: 1, 2, 3. So, we have a contradiction. Suppose now, that the geometry of a rotating plane has changed. However, what will be more preferable in such a case: the top or the bottom? The problem is symmetric, in fact; to what side the plane has curved in such a case? If we make the last supposition, that the radius has curved (as the apparent motion changes in the non-inertial system), then the second observer will see it as non-curved, whereas the ﬁrst and third observers will consider it as ”curved” to diﬀerent sides. Thus, three observers will see diﬀerent pictures at the same point for

78

CHAPTER 2. GRT BASIS

?

r

O

? ? 1 23

?

Figure 2.1: The geometry of a rotating circle.

2.2 CRITICISM OF GRT BASIS

79

Figure 2.2: Equidistant observers at a circle.
the same space; therefore, the curvature of the radius is not an objective fact (and cannot be a matter for scientiﬁc enquiry).
The rotating circle proves the contradictive nature of SRT and GRT ideas. Really, according to the textbooks, the radius, which is perpendicular to the motion, does not change. Therefore, the circles will remain at their places irrespective of the motion. Let us seat the observers on a motionless circle at equal distances from each other and produce a pointlike ﬂash from the center of a circle, in order the observers to draw the strokes on a moving circle at the time of signal arrival (Fig. 2.2). Owing to the symmetry of a problem, the strokes will also be equidistant. At subsequent periodic ﬂashes (with the appropriate period) each observer will conﬁrm, that a stroke mark passes by him at the ﬂash instant, that is, the lengths of segments of motionless and rotating circles are equal. When the circle stops, the marks will remain at their places. The number of equidistant marks will not change (it equals to the number of observers). Therefore, the lengths of segments will be equal in the mo-

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tionless case as well. Thus, no contraction of lengths (and change of

geometry) took place at all.

Now we consider again the space geometry problem, but with the

other approach. This problem is entirely confused still since the times

of Gauss, who wanted to determine the geometry with the help of light

beams. The limited nature of any experiment cannot inﬂuence the ideal

mathematical notions, does it? Note, that in GRT the light even moves

not along the have in GRT

[s1h7o]:rtδest(1p/a√thg:00in)dstle=ad0o, fwFheerrme agtα’βs

principle is metric

δ dl = 0, we tensor. What

does distinguish the light in such a case? The necessity of changing the

geometry is often ”substantiated” in textbooks as follows: in order the

light to ”draw” a closed triangle in the gravitational ﬁeld, the mirrors

should be turned around at some angle; as a result, the sum of angles of

a triangle will diﬀer from π. However, for any point-like body and three

reﬂectors in the ﬁeld of gravity (see Fig. 2.3) the sum of ”angles” can be

written as:

βi = π + 4 arctan

gL 2v02

− 2 arctan

gL v02

.

It occurs, that the geometry of one and the same space depends on the conditions of the experiment: on L and v0. Since the angle α between the mirrors A and B can also be changed (we chose α = 0 in our Fig. 2.3), we have a possibility of artiﬁcial changing the geometry within wide limits. Note, that the same variable parameters α and L remain for the light as well. In such ”plausible” proofs of the necessity of changing the geometry some important points are not emphasized. First, both in the experiment with material points, and in the experiment with the light the geometry is ”drawn” sequentially during some time, rather than instantaneously. Second, for accelerated systems the particles (and the light) move in vacuum rectilinearly, according to the law of inertia, and, actually, the motion of the boundaries of this accelerated system is imposed on this motion additively. All angles of incidence (in the laboratory system) are equal to corresponding angles of reﬂection, and the ”geometry of angles” does not change at all. Simply, the ﬁgure is obtained unclosed because of motion of the boundaries. Third, the role

2.2 CRITICISM OF GRT BASIS

A

L

81 B

g
C Figure 2.3: ”Geometry of a triangle”.
1
2 Figure 2.4: Drawing of the straight line in the gravitational ﬁeld.
of the boundaries is not uncovered at all in determining the relations between the lengths of real bodies. For example, if all points of a real body are subject to the eﬀect of identical accelerating force, then the mutual relation between lengths and angles (the ”geometry”) remains unchanged. If, however, only the boundaries are subject to acceleration, then all real changes of bodies’ size take place only at interaction with the boundaries. In any case the Euclidean straight lines can be drawn. For example, to draw the horizontal straight line in the gravitational ﬁeld we take two similar long rods (Fig. 2.4). At the middle of the ﬁrst rod we install a point-like support. As a result of bending of a rod, the upward-convex line is generated. Then we install two point-like supports for the second rod at the level of two lowered ends of the ﬁrst

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CHAPTER 2. GRT BASIS

rod. As a result of bending of the second rod, the downward-convex line is generated. The middle line between these two bonded rods determines the straight line.

The equivalence principle
Now we shall turn to the next important GRT notion – the equivalence of the gravitational ﬁeld to some system non-inertiality. In contrast to any non-inertial system, the gravitational ﬁeld possesses some unique property: all moving objects deﬂect in it toward a single center. If we generate two light beams between two ideal parallel mirrors and direct them perpendicular to mirrors, then in the inertial system these beams will move parallel to each other for inﬁnitely long time. A similar situation will take place at acceleration in the non-inertial system, if the mirrors are oriented perpendicular to the direction of acceleration. And, on the contrary, in the gravitational ﬁeld with similar orientation of mirrors the light beams will begin to approach each other (Fig. 2.5). And, if some eﬀect will happen to be measured during the observation, then, owing to a great value of light speed, the existence of namely the gravitational ﬁeld (rather than the non-inertiality) can also be identiﬁed. Obviously, the curvature of mirrors should not be taken into consideration, since, along with gravitational forces there exist also the other forces, which can retain the mutual conﬁguration of mirrors. The distinction of a spherical symmetry from planar one can be found for weak gravitational ﬁelds as well. The GRT conclusion on the possibility of excluding the gravitational ﬁeld for some inertial system during the whole observation time is wrong in the general case.
The equivalence principle of the gravitational ﬁeld and acceleration can be related to one spatial point only, i.e. it is unreal: it leaded to a false result for the light beam deﬂection in the gravitational ﬁeld, for example (only later Einstein corrected the coeﬃcient in two times). The equivalence principle of the inertial and gravitating mass can be rigorously formulated also for a separate body only (it is unreal for GRT, since GRT involves interdependence of the space-time and all bodies). Because of this, GRT does not physically proceed to any non-relativistic theory at all (but formally mathematically only). All relativistic linear

2.2 CRITICISM OF GRT BASIS

83

g

g

g

Figure 2.5: Rapprochement of parallel light beams in the gravitational ﬁeld.

84

CHAPTER 2. GRT BASIS

transformations can be related to empty space only, since real bodies (even simply as reference points) lead to nonlinear properties of the space. Then, phenomena diﬀerences with changing reference systems must be studied for the same point (in the space and time). But how can two diﬀerent observers be placed at one point? Therefore, the relativistic approach can possess the approximate model character only (without globality).
It is not any surprising thing, that the same physical value – a mass – can participate in diﬀerent phenomena: as a measure of inertia for any acting forces, including the gravitational one, and as a gravitating mass (for example, a moving charge produces both electric and magnetic ﬁelds). The question on the rigorous equality of inertial and gravitating masses is entirely artiﬁcial, since this equality depends on the choice of a numerical value of the gravitational constant γ. For example, expressions (laws) retain the same form in the case of proportionality mg = αmin, but the gravitational constant will be deﬁned as γ′ = α2γ. It is not necessary to search any mystics and to create pictures of curved space. The substitution of the same value for the inertial and gravitating mass is made not only for GRT, but for the Newton’s theory of gravitation as well. It is nothing more than an experimental fact (more precisely, the most simple choice of the value γ).
When one comes to the dependence of a form of equations on spacetime properties [37], there exists some speculation for this idea. The impression is given that we can change this space-time to check the dependence claimed. In fact, the Universe is only one (unique). GRT tries to add a complexity of the Universe to any local phenomena, which is not positive for science. The choice of local mathematical coordinates is a diﬀerent matter (a phenomenon symmetry can simplify the description in this case) and globality is not the case again.
The use of non-inertial systems in GRT is contradictory intrinsically. Really, in a rotating system rather distant objects will move at velocity greater than light speed; but SRT and GTR assert, that the apparent velocities should be lower, than c. However, the experimental fact is as follows: the photograph of the sky, taken from the rotating Earth, indicates, that the visible solid-state rotation is observed. The use of a

2.2 CRITICISM OF GRT BASIS

85

rotating system (the Earth, for example) does not contradict the classical physics at any distance to the object from the center, whereas in GRT the value of g00 component becomes negative, but this is inadmissible in this theory. What’s about astronomical observations (from the Earth)?

Time in GRT
The notion of time in GRT is confused beyond the limit as well. What does it mean by the clock synchronization, if it is possible only along the unclosed lines? The change of the moment of time reference point in moving around a closed path is an obvious contradiction of GRT, since at a great synchronization rate many similar passes-around can be made, and arbitrary aging or rejuvenation can be obtained. For example, considering the vacuum (emptiness) to be rotating (if we ourselves shall move around a circle), we can get various results depending on a mental idea.
If we momentarily believe the GRT dependence of time from the gravitational potential and believe the equivalence of gravitation and non-inertiality (an acceleration), then it could be easily understand that time depends on the relative acceleration in this case (it is an extended interpretation). Really, diﬀerent accelerations correspond to diﬀerent gravitational potentials in this case, and conversely. But relative accelerations possess the vector character (and it cannot be ”hidden”), that is the extended interpretation is the only possible one. Using the modiﬁed paradox of twins [51], the independence of time on acceleration for extended interpretation can easily be proven. Let two astronauts – the twins – are at a great distance from each other. On a signal of the beacon, situated at the middle, these astronauts begin to ﬂy toward a beacon at the same acceleration (Fig. 2.6). Since in GRT the time depends on the acceleration and the acceleration has relative character, each of the astronauts will believe, that his twin brother is younger than he is. At meeting near the beacon they can exchange photos. However, owing to the problem symmetry, the result is obvious: the time in an accelerated system ﬂows at the same rate, as in non-accelerated one. Besides, each astronaut (third observer can be placed at the beacon) can send the signals to the other one about his each birthday. The same number of

86
a

CHAPTER 2. GRT BASIS
a

Figure 2.6: The ﬂy of twins with an acceleration.
light spheres will be perceived by each astronaut till they meet at the beacon (there is nowhere to hide the spheres). Having received a ”telegram” about 50th birthday of the brother a minute before the meeting, whether the other astronaut will congratulate the brother on his 5th birthday (maybe, he needs the oculist)? If we suppose the gravitational ﬁeld to be equivalent to the acceleration (according to GRT), then we obtain, that the time intervals do not depend on the gravitational ﬁeld presence. For example, the extend interpretation which includes the relationship between time and acceleration can be easily disproved in the following manner. Let us consider several mans in diﬀerent parts of the Earth. If we will use the GRT equivalence of the gravitational ﬁeld and an acceleration, then, to imitate the terrestrial attraction, they must be accelerated from the Earth’s center, that is in diﬀerent directions (all acceleration vectors will diﬀer their directions). Therefore, all relative accelerations will be diﬀerent. Owing to the problem symmetry, the result is obvious: the age of these mans will be independent on their location.
Now we make some remarks concerning the method of synchronization of times by means of a remote periodic source disposed perpendicular to the motion of a body [48]. We begin with inertial systems. The possibility of time synchronization on restricted segments of the trajectory makes it possible to synchronize the time throughout the line of motion (Fig. 2.7). Indeed, if for each segment there exists an arbitrarily remote periodic source Nj sending the following information: its number Nj, the quantity nj of passed seconds (the time reference point is not coordinated with other sources), then the observers at junctions of segments can compare the time reference point for a source on the left and for a source on the right. Transmitting this information sequentially

2.2 CRITICISM OF GRT BASIS

87

N1

N2

N3

Nn

L

L

L

1

2

3

Ln

Figure 2.7: The time synchronization throughout the line of motion.

from the ﬁrst observer to the last one, it is possible to establish a single time reference point (the time itself, as it was shown in Chapter 1, has absolute sense [48]).
Apparently, the observed rate of transmission of synchronization signals has no eﬀect on the determination of duration of times: the pulses (for example, light spheres or particles), which mark the number of passed seconds, will equidistantly ﬁll the whole space, and the number of spheres emitted by a source will be equal to the number of spheres, which reach the receiving observer. (We are not the gods, you see, to be able to introduce the ”beginning of times”: the time takes already its normal course and elapses uniformly.) Even if we consider the apparent signal propagation rate to be c = c(r), then, irrespective of the path of light, the number of spheres reached the receiving observer (having a zero velocity component in the source direction) will be the same as the number of spheres emitted by a source (simply, the spheres can be spatially thickened or rareﬁed somewhere). Time as the duration will be perceived uniformly. Thus, the full synchronization is possible in the presence of spatial inhomogeneities (of the gravitational ﬁeld) as well.

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CHAPTER 2. GRT BASIS

We would remind two well-known experiments which were urgently ascribed by relativists to GRT advantages. The Hafele-Keating’s experiment consisted in the following: two pairs of cesium atomic watch ﬂew at an airplane in the east and west directions, and their readings were compared with the resting watch (in so doing the SRT ”velocity eﬀect” was taken into consideration, but its lack was proved in Chapter 1 of the present book). The Pound-Rebka’s experiment consisted in the following: using the Mossbauer eﬀect, a frequency shift was detected for a photon which passed some distances in the vertical directions (both up and down). In physics it is not accepted to take into account the same eﬀect twice. It is clear, that the acceleration and gravitation express some force, that inﬂuences various processes. But this will be the general result of the eﬀect of namely the forces. For example, not any overload can be withstood by a man, the pendulum clock will not operate under zero gravity, but this does not mean, that the time stopped. Therefore, the rough Hafele-Keating’s experiment states the trivial fact, that the gravitation and acceleration somehow inﬂuence the processes in a cesium atomic watch, and the high relative accuracy of this watch for a ﬁxed site is fully groundless. Besides, interpretation of this experiment contradicts the ”explanation” of the Pound-Rebka’s experiment with supposition about independence of frequency of emission in ”the units of intrinsic atom time” [3] on gravitational ﬁeld. Besides, a further uncertainty in GRT must be taken into consideration: there can exist immeasurable rapid ﬁeld ﬂuctuations (with a rate greater than inertness of measuring instruments) even in the absence of the mean ﬁeld g. Such the uncertainty exists for any value of g: since the time in GRT depends on the gravitational potential, then an eﬀective potential will be nonzero even with < g >= 0. Whether is it possible to invent, though theoretically, a precise watch, which can be worn by anybody? Probably, a rotating ﬂywheel with a mark (in the absence of friction – on a superconducting suspension), whose axis is directed along the gravitational ﬁeld gradient (or along the resultant force for non-inertial systems) could read out the correct time. At least, no obvious reasons and mechanisms of changing the rotation rate are seen in this case. Certainly, for weak gravitation ﬁelds such a watch will be less accurate

2.2 CRITICISM OF GRT BASIS

89

at the modern stage, than cesium one. Outside the criticism of relativity theory, we hypothesize, that atom decay is anisotropic, and this anisotropy can be interrelated with a direction of the atomic magnetic moment. In this case we can regulate atomic moments and freeze the system. Then, the ”frozen clock” will register diﬀerent time depending on its orientation in the gravitational ﬁeld.
Now we return to synchronizing signals (for simultaneous measurement of lengths, for example). For a rectilinearly moving, accelerated system it is possible to use the signals from a remote source being perpendicular to the line of motion, and for the segment of a circle the source can be at its center. These cases actually cover all non-inertial motions without gravitation. (Besides, for the arbitrary planar motion it is possible to make use of a remote periodic source being on a perpendicular to the plane of motion.) For the real gravitational ﬁeld of spherical bodies in arbitrary motion along the equipotential surfaces it is possible to use periodic signals issuing from the gravitational ﬁeld center.
Note, that to prove the inconsistency of SRT and GRT conclusions on the change of lengths and time intervals it is suﬃcient, that the accuracy of ideal (classical) measurement of these values could principally exceed the value of the eﬀect predicted by SRT and GRT. For example, for a synchronizing source being at the middle perpendicular to the line of motion we have for the precision of the time of synchronization: ∆t ≈ l2/(8Rc), where l is the length of a segments with the synchronized time, R is the distance to the synchronizing source; that is, ∆t can be decreased not only by choosing the great radius of a light sphere, but also by choosing a small section of motion l. From the SRT formulas on time contraction we have for the similar value: ∆t = l(1 − 1 − v2/c2)/v. If for ﬁnite R and speciﬁed speed v we choose such l, that the inequality

l/(8Rc) < (1 − 1 − v2/c2)/v,

(2.1)

be met, then the conclusions of relativistic theories occur to be invalid. For the system arbitrarily moving along the radius (drawn from the
gravitational ﬁeld center) it is possible to use for synchronization a free falling periodic source on the perpendicular to the line of motion. In

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CHAPTER 2. GRT BASIS

this case R should be chosen of such value, that the ﬁeld cannot actually change (due to equipotential sphere rounding) at this distance, and corresponding l from (2.1) near the point, to which the perpendicular is drawn. Therefore, the GRT conclusions can be refuted in this case as well. For the most important special cases the ”universal” SRT and GRT conclusions on the contraction of distances as a property of the space itself are invalid. In the most general case it seems intuitively quite obvious, that such a position of a periodic source can be found, that the signal to come perpendicular to the motion, and that such R and l from (2.1) to exist, which refute the GRT results. There is no necessity at all in a ”spread” frame of reference and in an arbitrarily operating clock: any change of real lengths should be explained by real forces; it is always possible to introduce a system of mutually motionless bodies and the universal time (even if it were the recalculation method). Thus, the space and time must be Newtonian and independent on the motion of a system.

Some GRT corollaries
Now we pass to mathematical methods of GRT and to corollaries of this theory. The games with the space-time properties result in the fact, that in GRT the application of variation methods occurs to be questionable: the quantities are not additive, the Lorentz transformations are noncommutative, the integral quantities depend on the path of integration. Even it is not clear, how the terminal points can be considered as ﬁxed, if the distances are diﬀerent in diﬀerent frames of reference.
Because of nonlocalizableness (non-shieldness) of gravitation ﬁeld, conditions on inﬁnity (because of the mass absence on inﬁnity, it is euclideanness) are principally important for the existence of the conservation laws in GRT [37] (for systems of the insular type only). The classical approach is more successive and useful (theoretically and practically): energy is determined correctly to a constant, since the local energy diﬀerence between two transition points has a physical meaning only. Therefore, conditions on inﬁnity is groundless.
Highly doubtful is the procedure of linearization in the general form, since it can be only individual. The tending to simplicity is declared, but

2.2 CRITICISM OF GRT BASIS

91

even two types of time are introduced – coordinate and intrinsic time. The ﬁtting to the well-known or intuitive (classically) result is often made. So, for motion of Mercury’s perihelion [3] the du/dϕ derivative can have two signs. Which of them should be chosen? To say already nothing of the fact, that the dividing by du/dϕ is performed, but this quantity can be zero. The complexity of spatial-temporal links is stated, but eventually one passes for a very long time to customary mathematical coordinates; otherwise there is nothing to compare the results with. For what was there a scrambling? For pseudo-scienceness?
Till now there is no suﬃcient experimental proof of whether the rate of transmission of gravitational interactions is higher than, lower than, or exactly equal to, light speed (as is postulated in GRT). For example, on the basis of observations, Laplace and Poincare believed [24,87] that the rate of transmission of gravitational interactions is several orders greater than the light speed.
Now we note on the experimental substantiation of the GRT. Usually, even there exists a hundred diﬀerent data, a theory is constructed not always: the data can simply be tabulated in a table. But in the case of the GRT we see ”the Great theory of three and half observations”, three of which are the ﬁction. Concerning the light deﬂection from rectilinear motion in a gravitational ﬁeld, we should make the following statements. First, as it was pointed out by many experimentalists, a quantitative veriﬁcation of an eﬀect essentially depends on the faith of the concrete experimentalist. Second, even from the classical formula ma = γmM r/r3 it follows that any ”object” (even of zero or negative mass) will ”fall down” in the gravitational ﬁeld. Third, with which a value does the eﬀect be compared? With a value in empty space? As early as 1962, a group of Royal astronomers declares that the light deﬂection near the Sun cannot be considered as conﬁrmation of GRT, because the Sun has an atmosphere stretching for a great distance. We would remind that the eﬀect of refraction is long taken into account by astronomers for the terrestrial atmosphere. Lomonosov discovers the deﬂection of a light beam in the atmosphere of the Venus long ago. For explanation, imagine a glass sphere. Naturally, parallel rays (from distant stars) will be deﬂected to the center in it. Such a system is well

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CHAPTER 2. GRT BASIS

known as an optical lens. The similar situation will take place for a gas sphere (the Sun’s atmosphere). For accurate calculation of light beam deﬂection in the gravitational ﬁeld, one should take into account that the presence of the solar atmosphere and the fact, that the presence of density and temperature gradients on the beam path causes changes to the medium’s refractive index and, hence, to the bending of the light beam. Even at the distance of a hundred of meters (near the Earth), these eﬀects cause a mirage, so ignoring them for a beam coming from a star and passing near the Sun (at millions of kilometers) is a pure speculation.
The displacement of the perihelion of Mercury is, of course, a remarkable eﬀect, but whether the sole example is insuﬃcient ”to attract” a scientiﬁc theory, or not? Therefore, it would be interesting to observe it near solid bodies (for the satellite of the planets, for instance), so that the value of this eﬀect could be estimated for certainty. The matter is that the Sun is not a solid body, and the motion of Mercury may cause a tidal wave on the Sun, which may in turn also cause a displacement of Mercury’s perihelion. (Depending on the rate of transmission of gravitational interactions and ”hydrodynamic” properties of the Sun, the tidal wave may either outstrip, or lag behind the motion of Mercury.) In any case, it is necessary to know the rate of transmission of gravitational interactions for calculating the eﬀect of a tide due to the Mercury and other planets on Mercury’s orbit characteristics, in order that the purely ”gravitational” eﬀect (if it exists) of the general relativity theory could be separated.
Calculating the perihelion displacement in GRT (from the rigorous solution for a single attractive point), the impression is given that we know astronomical masses exactly. If we use GRT as a correction to Newton’s theory, the situation is in fact opposite: there exists a problem knowing visible planet motions to reestablish the exact planet masses (to substitute the latters and to check GRT thereafter). Imagine the circular planet orbit. It is obvious in this case, that the Newtonian rotation period will already be taken with regard to an invisible precession, i.e. the period will be renormalized. Therefore, renormalized masses of planets are already included in Newton’s gravitation theory. Since

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the GRT-corrections are much less than the perturbation planet actions and the inﬂuence of a non-sphericity, the reestablishment of exact masses can essentially change the description of a picture of the motion for this complex many-body problem. No such detailed analysis was carried out. Generally speaking, the situation with description of the displacement of the Mercury’s perihelion is typical for relativist’s behaviour. First, it was declared that the eﬀect was predicted, but Einstein compares it with the well known results of approximate calculations, which was produced by Laplace long before origin of the GRT. Hope, each man understands a great diﬀerence between ”predict” and ”explain after the event” (remember the appropriate anecdote of Feynman). Second, there exists the most part of precession already in classical physics: the data of 19th century was found with taking into account inﬂuences of some planets. The result obtained was the value of 588”, whereas a deﬁciency in the calculated value make up about 43” only, that is a small correction. (Note, that some data of 20th century indicate the total value of precession to be about 10 times higher than mentioned one, but the ”deﬁciency for GRT” in 43” is maintained - ”taboo”; nevetheless, it could be a misprint and we will not cavil to 1/3 of ”the great experimental base of GRT”). Third, the exact calculation for a many-body problem cannot yet be made even by the modern mathematics. In classical case the calculation was made as a sum of independent corrections from inﬂuences of separate planets (the Sun and planets were considered as material points). Naturally, the classical net result (more than 90 % from observable one!) can some more be improved with taking into account the solar non-sphericity, inﬂuences of all planets (including small bodies) of the solar system, the fact that the Sun is not a solid object (a material point) and its local density in diﬀerent layers must ”follow” inﬂuences of other moving planets. Most probably, this way of using real physical mechanisms can lead to obtaining the deﬁcient small eﬀect. But the relativist’s declaration is inconceivable speculation! They ”found” an eﬀect (the small procent only) considering motions of two material points only - the Sun and the Mercury. Sorry, and what will a correction be made with the GRT for the most part of the eﬀect obtained classically? Do you fear to calculate? Then on what ”a brilliant coincidence”

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O

A

BC

Figure 2.8: The fall on a ”black hole”.
do you repeat? It is the pure machination to a desired result!
The prototype of the ”black hole” in Laplac’s solution, where the light, moving parallel to the surface, begins to move over a circle like the artiﬁcial satellite of the Earth, diﬀers from the GRT ideas. Nothing prohibits the light with a rather high energy to escape the body in the direction perpendicular to its surface. There is no doubt, that such beams will exist (both by internal and external reasons): for example, the beams falling from outside will be able to accumulate energy, in accordance with the energy conservation law, and to leave such a ”black hole” after reﬂecting. Instead of invoking contradictory properties of light, we simply consider the ”fall” of an elementary particle – an electron, for example. Whether the possibility of the elastic reﬂection is maintained for it, or such the possibility must postulatively be forbidden (to rescue the GRT)? And if such the possibility is not forbidden, then we consider the following process. Let an electron be coming into fall with the zero start velocity from a distant point A (at the distance 100 a.u., for example) to a very massive body (Fig. 2.8). The body absorbs ”last surplus nearest molecules” and becomes the ”black hole” in a matter of an instant before the electron crosses the Schwarzschild sphere (which is marked as B on the picture). To be visual, the distance |OB| is shown comparatively large. In a matter of an instant before the collision of the electron with the surface of the ”black hole” the latter

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object was stable, and since neither velocity nor acceleration of this surface can instantly become very large (besides, the collision can take place with a particle ﬂying to meet), then at the elastic collision the electron will ﬂy to the point A with just the same speed as it acquires before the collision. Relativists claim that it cannot get over the Schwarzschild sphere B. Let it come to a stop at the point C (at the distance 10 km from the body center, for example). If the energy conservation law is obeyed, and since the electron’s velocity equals to zero at the points A and C, then the potential energy of the electron at the point A is equal to the potential energy at the point C. Therefore, the gravitational ﬁeld (attractive forces) is absent between the points A and C, or else the potential might be monotonically decreasing. However, the consideration of the situation from the pure GRT positions leads to the still worse result (see below). The ”black holes” in GRT is a real mysticism. If we take a long rod, then at motion its mass will increase and the size will decrease (according to SRT). What will happen? Is the ”black hole” generated? All the sky will become ﬁlled with ”black holes”, if we shall move rapidly enough. And, you see, this process would be irreversible in GRT. For example, any object of the Universe is a ”black hole” for fast moving light (how it can exist?).
Recall some well-known solutions: 1) the Schwarzschild solution describes the centrally symmetric ”ﬁeld” in vacuum (note that the temperature characteristic is absent, i.e. T = 0K); and 2) the axially symmetric Kerr’s metric describes the ”ﬁeld” of a rotating collapsing body. The presence of singularities or multiple connection of the solution implies, that, as a minimum, the solution is inapplicable in these regions. Such a situation takes place with the change of the space - time signature for the ”black hole” in the Schwarzschild solution, and it is not necessary to search any artiﬁcial philosophical sense in this situation. The singularity in the Schwarzschild solution for r = rg cannot be eliminated by purely mathematical manipulations: the addition of the inﬁnity with the other sign at this point is the artiﬁcial game with the inﬁnities, but such a procedure requires the physical basis. (You see, all singularities at zero are not eliminated by artiﬁcial addition of α exp (−λr)/r, where λ is a large quantity).

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Even from GRT follows the impossibility of observation of ”black holes”: the time of the ”black hole” formation will be inﬁnite for us as remote observers. Even if we were waited till ”the End of the World”, no one ”black hole” could have time to form. And since the collapse cannot be completed, the solutions, which consider all things as though they have already happened, have no sense. The separation of events by inﬁnite time for internal and external observers is not ”an extreme example of the relativity of the time course”, but the elementary manifestation of the inconsistency of Schwarzschild’s solution. The same fact follows from the ”incompleteness” of systems of solutions. It is not clear, what will happen with the charge conservation law, if a greater quantity of charges of the same sign will enter the ”black hole”? The mystical description of ”metrical tidal forces” [39] at approaching the ”black hole” is invalid, since it would mean, that the gravitation force gradient is great within the limits of a body, but all GRT ideas are based on the opposite assumptions. The Kerr metric in the presence of rotation also clearly demonstrates the inconsistency of GRT: it gives in a strict mathematical manner several physically unreal solutions (the same operations, as for Schwarzschild’s metric, do not save the situation). Thus, such the GRT objects as the ”black holes” cannot exist and they must be transfered from the realm of sciences to the province of the non-scientiﬁc ﬁction. All the Universe is evidence of the wonderful (frequently dynamical) stability: there do not exist inﬁnite collapses (an explosion can happen sooner). All this does not cancel a possibility of the existence of superheavy (but dynamically stable) objects which can really be manifested by several eﬀects (for example, by accretion, radiation etc.). No the GRT fabrications are required for these purposes at all. We have no need to seek ways for the artiﬁcial rescue of the GRT, such as the ”evaporation of the black holes”, since such a possibility is strictly absent in the GRT (the speed of light cannot be overcome). On the contrary, in classical physics no problems exist at all.
GRT contains a lot of doubtful prerequisites and results. List some of them. For example, the requirement of gravitational ﬁeld weakness for low velocities is doubtful: if the spacecraft is landed on a massive planet, whether it can not stand or slowly move? Whether some molecules

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with low velocities cannot be found in spite of temperature ﬂuctuations? The consideration of a centrally symmetric ﬁeld in GRT has not physical sense as well: since the velocity can be only radial, then not only rotations, but even real temperature characteristics can not exist (i.e. T = 0 K). The ﬁeld in a cavity is not obtained in a single manner, but, simply, two various constants are postulated in order to avoid singularities.
The emission of gravitation waves for a parabolic motion (with eccentricity e = 1) results in the inﬁnite loss of energy and angular momentum, which obviously contradicts the experimental data.
In fact, GRT can be applied only for weak ﬁelds and weak rotations, i.e. in the same region, as the Newtonian theory of gravitation. Recall that the similar interaction between moving charges diﬀers from the static Coulomb law. Therefore, prior to applying the static Newtonian law of gravitation, it must be veriﬁed for moving bodies, but this is a prerogative of the experiment.
Let us discuss one more principal point concerning the relativity of all quantities in GRT. The laws, written simply as the equations, determine nothing by themselves. The solution of any problem still requires the knowledge of speciﬁc things, such as the characteristics of a body (mass, shape etc.), the initial and/or boundary conditions, the characteristics of forces (magnitude, direction, points of application etc.). The ”reference points” are actually speciﬁed, with respect to which the subsequent changes of quantities (position, velocity, acceleration etc.) are investigated. The principal relativity of all quantities in GRT contradicts the experiments. The subsequent artiﬁcial attempt to derive accelerations (or rotations) with respect to the local geodesic inertial Lorentzian system – this is simply the ﬁtting to only workable and experimentally veriﬁed coordinates of the absolute space (GRT does not contain any similar things organically [18]).
The gravitational constant is not a mathematical constant at all, but it can undergo some variations [9]. Therefore, this value can account corrections to Newton’s static law of gravitation (for example, these inﬂuences do not taken into consideration for the displacement of the perihelion of the Mercury). We are reminded that in ﬁnite moving

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(periodic, for example) diﬀerent resonance phenomena can be observed for a coupled many-body system. The eﬀect is manifested in a conforming correction of orbital parameters (especially taking into account a ﬁnite size of bodies: non-sphericalness of their form and/or of the mass distribution).

Generally speaking, the theory of short range for gravitation could be useful (but it can be not useful depending on the gravitation transmission rate) for the ﬁnite number of cases only: for the rapid (v → c) motion of massive (the same order) bodies close to each other. The author does not know such practical examples.

The GRT approach to gravitation is unique: to be shut in the lift (to take pleasure from the fall) and to be not aware that the end (hurt oneself) will be after a moment. Of course, the real state is quite diﬀerent one: we see always where and how we move relative to the attractive center. Contrary to Taylor and Wheeler, it is the second ”particle”, together with the ﬁrst ”particle” – with the observer. That is the reason that the pure geometric approach is a temporal zigzag for physics (although it could ever be useful as a auxiliary technique). And two travelers from the parable [33] (allegedly demonstrate the approach of the geometry of curved space) have need for ”very little”: for the wish to move from the equator just along meridians (on the spheric earth surface), but the rest of ﬁve billion mans can not have such the wish. Contrary to traveler’s wish, the wish ”to do not attract to the Earth (or the Sun) and to ﬂy away to space” is inadequate. The notion force (the force of gravity in this case) reﬂects this fact. Geometry cannot answer to the following questions: how many types of interactions exist in nature, why there exist they only, why there exist local masses, charges, particles, why the gravitational force is proportional just to r2, why there realize the speciﬁc values of physical constants in nature, and many other questions. These problems are the physical (experimental) prerogative.

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2.3 Criticism of the relativistic cosmology

The theories of evolution of the Universe will remain the hypotheses for ever, because none of assumptions (even on the isotropy and homogeneity) can be veriﬁed: ”a moving train, which departed long ago, can be catched up only at the other place and at the other time”. GRT assigns to itself the resolution of a series of paradoxes (gravitational, photometric, etc.). Recall that the gravitational paradox consists in the following: it is impossible to obtain the deﬁnite value of the gravitational acceleration of a body from Poisson’s equation for the inﬁnite Universe possessing a uniform density. (What relationship to the reality bore pure mathematical uncertainties with conditions on inﬁnity for a physical model?) Recall also the essence of the photometric paradox: for the inﬁnitely existing (stationary) inﬁnite Universe the brightness of sky must be equal to the mean brightness of stars without considering the light absorption and transform (again we have rather many unreal assumptions). However, the classical physics has also described the possibilities of resolution of similar paradoxes (for example, by means of systems of diﬀerent orders: Emden’s spheres, Charlier’s structures, etc.). Apparently, the Universe is not a spread medium, and we do not know at all its structure as a whole to assert the possibility of realization of conditions for similar paradoxes (more probably, the opposite situation is true). For example, the Olbers photometric paradox can easily be understood on the basis of the analogy with the ocean: the light is absorbed, scattered and reﬂected by portions, and the light simply ceases to penetrate to a particular depth. Certainly, such a ”depth” is huge for the rareﬁed Universe. However, the ﬂashing stars represent rather compact objects spaced at great distances from each other. As a result, only a ﬁnite number of stars make a contribution into the light intensity of the night sky (to say nothing of the fact that the Doppler eﬀect, or, more better, the red shift as the experimental fact, must be also taken into account in the theory).
The situation with the red shift in spectra of astronomical objects does not be ﬁnally clariﬁed. In the Universe there exists a considerable number of objects with quite diﬀerent shifts in diﬀerent spectral