AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH

© 2014, Science Huβ, http://www.scihub.org/AJSIR

ISSN: 2153-649X, doi:10.5251/ajsir.2014.5.2.40.52 Correction of the special theory of relativity:

physical reality and nature of imaginary and complex numbers

Alexander A. Antonov1*

1Research Center of Information Technologies “TELAN Electronics” *Correspondence author: tel/fax (380) 444243587; E-mail: telan@bk.ru

ABSTRACT

It is demonstrated that regardless of the results of the MINOS, OPERA and ICARUS experiments, the current interpretation of the special theory of relativity (STR) has been refuted by the theoretical and experimental investigation of processes in linear electric circuits that allowed proving the general scientific principle of physical reality of imaginary and complex numbers. The physical reality of imaginary and complex numbers is explained. The adjusted interpretation of the STR is suggested, where formulae describing the relativistic effect are corrected accordingly. The new hypothesis of the Multiverse is suggested. Contrary to all other Multiverse hypotheses, it is real, i.e., it can be verified by people visiting other parallel Universes.

Keywords: imaginary numbers, special theory of relativity, hidden dimensions, Multiverse, parallel Universes;

INTRODUCTION

Any science is based on a certain set of fundamentals; at least some of them are accepted on faith, others are borrowed from other sciences. In mathematics, the former are referred to as axioms, in other sciences – as dogmata. Contrary to axioms, dogmata eventually become obsolete and are refuted.

For instance, in the special theory of relativity (STR), these dogmata have it, in particular, that physical bodies cannot move at superluminal speed, whereupon imaginary (this is where their name stems from) and complex numbers have no physical meaning.

It is noteworthy that imaginary numbers were discovered in mathematics over 500 years ago (Vinogradov, 1979), i.e. long before the STR. Over these years, mathematicians failed to reveal the physical meaning of imaginary numbers. Engineers faced the same problem almost 200 years ago after the phenomenon of resonance was discovered (Blanchard, 1941). However, both mathematicians and engineers, having failed to solve the problem, left open the question on the physical meaning of imaginary numbers.

When it came to physicists, they decided to close the issue by alleging that imaginary numbers have no physical meaning. However, to some this conventional interpretation of the STR did not seem convincing enough (Goldsmith, 1997; Marchal, 1999; Kalinin, 2003; Artekha, 2007; Sekerin, 2007; Albert

and Galchen, 2009; Petrov, 2012), and already in the 21

st century physicists published the results of the

MINOS and OPERA experiments, where superluminal neutrino were supposedly registered. However, the physical community believed these experiments were not trustworthy enough.

Theoretical proof of physical reality of imaginary and complex numbers: Nevertheless, regardless of the results of the MINOS, OPERA and ICARUS experiments, in early 21

st century the results of other

theoretical and experimental investigations were published, offering conclusive evidence disproving the dogma of the STR discussed herein (Antonov, 2013a,b,c).

These investigations, contrary to the MINOS, OPERA and ICARUS experiments, belong not to the elementary particle physics, but to the physics of oscillation processes in linear electric circuits. Thus, they can be experimentally verified in any radio-electronic laboratory. Nevertheless, the principle of physical reality of imaginary and complex numbers is true for physics in general, including the STR.

Let us consider evidences (their two).

It is well known that processes in linear electric circuits are described with a differential equation (or systems of these equations)

ya...dt

yda

dt

yda 01n

1n

1nn

n

n

Am. J. Sci. Ind. Res., 2014, 5(2): 40-52

41

xb...dt

xdb

dt

xdb 01m

1m

1mm

m

m

(1)

where )t(x is the input action (or the input signal);

)t(y is the response (or the output signal);

01mm01nn b,...b,b,a,...a,a are constant

coefficients;

0,...1m,m,0,...1n,n

is the order of

derivatives.

The solution of equation (1), as is well known, has two components

freeforc )t(y)t(y)t(y (2)

where forc)t(y is the forced component of

response;

free)t(y is its free (or transient) component.

Investigation of any of these components allows proving the physical reality of imaginary numbers, however, in a different way. Let us demonstrate it.

Let us start with the evidence based on investigation of the free component of response (Antonov, 2010a,b). When transient processes are studied, their particular type is found by solving the so-called characteristic algebraic equation, which corresponds to the initial differential equation (1)

0a...papa 01n

1nn

n (3)

where 01nn a,...a,a are the same as in Eq. (1)

constant coefficients;

0,1,...2n,1n,n are the exponential

quantities equal to the order of the corresponding derivatives in the differential equation (1); p is a variable which, in case it takes values in

the form of complex numbers i , is often

referred to as complex frequency;

1i is the imaginary unit.

In textbooks on the linear electric circuit theory, characteristic algebraic equations are always solved only on the set of complex numbers, contrary to mathematics, where algebraic equations can be solved both on the set of real and complex numbers.

This is a very interesting circumstance which deserves special discussion.

The matter is that on the set of real numbers algebraic equations of the n -degree, depending on

the values of their coefficients 01nn a,...a,a , can

have n solutions, 1n solutions, 2n solutions

or even 0 solutions. On the set of complex numbers

algebraic equations of the n -degree always have n

solutions at any values of the coefficients

01nn a,...a,a . Consequently, at certain values of

their coefficients 01nn a,...a,a algebraic equations

of an even n -degree can have none solutions on the

set of real numbers and have n solutions on the set

of complex numbers. As can be seen, one obviously excludes the other.

So, which of these algorithms gives the only correct result? Mathematicians are still unable to make their choice. However, engineers have known the answer to this question for a long time. The matter is that at

any combination of coefficients 01nn a,...a,a , a

certain transient process – aperiodic, critical or oscillation – is always observed in second-order electric circuits. In electric circuits of a higher order, a certain combination of these processes is observed. At that, the aperiodic transient process in the second-order electric circuit is observed if the solutions of the characteristic equation are real and different numbers. The critical transient process in the second-order electric circuit is observed if the solutions of the characteristic equation are real and equal numbers. The oscillation transient process is the second-order electric circuit is observed if the solutions of the characteristic equation are complex and different numbers, or complex-conjugate numbers, to be more precise.

However, if characteristic equations were solved on the set of real numbers, oscillation transient processes in nature were not supposed to exist, because they have no corresponding solution in the form of real numbers. Then, in particular, shock oscillations would not exist, including tsunami, the sound of a piano, a kid’s swing swinging after it is pushed.

Therefore, the solutions of characteristic (and not only characteristic) algebraic equations only the set of complex numbers, as well as complex numbers themselves, must be recognized as physically real.

Similar results are obtained after investigating the forced component of the solution of the differential equation (1), or, to be more precise, of the forced

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42

oscillations resonance (Antonov, 1987; 2008; 2009; 2010c). It is well known that resonance in electric circuits has three attributes:

1. At the resonance frequency, the immittance function (e.g., complex impedance) takes an extreme absolute value;

2. At the same resonance frequency (because

one and the same phenomenon – resona– is under investigation) the imaginary component of the immittance function (e.g., complex impedance) takes zero value;

3. Resonance frequency equals the frequency of free oscillations (e.g., of the shock oscillations discussed above).

Fig 1. Resonance electric LCR-networks which are series-oscillatory circuits

However, it turned out that resonance has these attributes only in electric LC-circuits. In electric LCR-circuits resonance hardly ever has these characteristics, if analysed precisely in terms of mathematics.

Indeed, precise analysis of series-oscillatory LCR-circuits (Antonov, 2008) shown in figure 1, allows finding the following frequencies.

For the LCR-circuit shown in figure 1a, these will be the frequencies:

01res , (4a)

which corresponds to the first attribute of resonance;

02res , (4b)

which corresponds to the second attribute of resonance;

0

2

0

20

20free

Q2

1Q4

, (4c)

which corresponds to the third attribute of resonance;

where LC

10 ;

L

R2 0 ;

0

0

2Q

;

res is resonance frequency of forced

oscillations;

free is the frequency of free oscillations.

For the LCR-circuit shown in figure 1b, these will be the frequencies:

0

2

020

201res

1res

Q

1Q4

0

(5a)

which corresponds to the first attribute of resonance;

0

Q

12QQ

48

2res

0

2

0

20

20

2002res

(5b)

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43

which corresponds to the second attribute of resonance;

0

2

020

20free

Q2

1Q4

, (5c)

which corresponds to the third attribute of resonance;

where LC

10 ,

RC

12 0 ,

0

0

2Q

.

For the LCR-circuit shown in figure 1c, the frequencies:

02

2

020

20

20

01res

1res

1Q

Q

4

(6a)

which corresponds to the first attribute of resonance;

2res

024

223

0

40

20

20

40

20

20

20

20

30

02res

1Q2Q

Q2QQ

168

48

(6b)

which corresponds to the second attribute of resonance;

0

2

020

20free

Q2

1Q4

, (6c)

which corresponds to the third attribute of resonance;

where LC

10 ,

RC

12 0 ,

0

0

2Q

.

As can be seen, one and the same physical phenomenon – resonance – in LCR-circuits is manifested through resonance frequencies which are different for different attributes of resonance. Moreover, in most cases, there turned to be two resonance frequencies corresponding to the same attribute of resonance. And all these resonance frequencies are not equal to the frequency of free oscillations.

Similar results are also obtained for any other type of electric LCR-circuits.

Since the results which correspond to the current interpretation of resonance at real frequencies have not been explained, all textbooks on electric circuit theory still resort to the explanation of resonance in electric LCR-circuits suggested in 1905 by K. A. Krug (1946), professor at Moscow Higher Technical College, which uses approximate formulae disguising the problem. However, the problem of inconsistency of the current interpretation of resonance still remains, and requires a solution.

Although, the difference of resonance frequencies

res corresponding to formulae (4), (5), (6) and the

frequency of free oscillations св from 0 is quite

insignificant and usually does not exceed the experimental error. Therefore, many scientists do not even suspect it exists. Nevertheless, there is a difference, and it has to be accounted for, just like the results of the MINOS, OPERA and ICARUS experiments, dealing with the same problem – the investigation of physical reality of complex numbers – had to be explained.

The author has developed the theory of resonance at complex frequencies free from the inconsistencies inherent to the theory of resonance at real frequencies. The new theory uses Cassini ovals (Antonov, 2008; 2009) and vector diagrams at complex frequencies (Antonov, 2009) in its reasoning.

In accordance with the theory, resonance in observed in the devices under investigation if complex input

frequency inpp and the frequency of free oscillations

freep converge, which proves the physical reality of

resonance at complex frequencies and of the complex frequencies themselves.

Thus, investigation of resonance at complex frequencies allowed proving the physical reality of imaginary and complex numbers, and, this, disprove the aforementioned dogma of the STR.

EXPERIMENTAL PROOF OF PHYSICAL REALITY OF IMAGINARY AND COMPLEX NUMBERS

However, some statements, which follow from the theory of resonance at complex frequencies (Antonov, 1987; 2008; 2009; 2010c), are so extraordinary that they are hard to believe without experimental evidence, whatever consistent the

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44

corresponding theoretical evidence and explanations might be. For example, here belongs the statement on resonance in the oscillatory LCR-circuit under the influence of exponential radio pulses, or the statement on resonance in aperiodic RC-circuits and RL-circuits under the influence of exponential video pulses.

However, the experiments described below (their four) speak for themselves. They are so simple that any engineer can verify them, and make sure that the conventional description of resonance oscillation processes is approximate and incomplete. This is why it had to be updated during the natural course of cognition.

Before describing particular experiments, it is necessary to note an important circumstance, which must be taken into account when performing them. The matter is that resonance is a regularity which applies only to the forced component of response to the input action. However, in electric circuits, under the influence of a signal, not only the forced component, but the free component of response is usually registered, as well, and it interferes with the observation of the forced component.

In passive electric circuits transient processes are always damped, and, therefore, after a while they fade. For this reason, they do not interfere into observation of forced sustained sinusoidal oscillations (as their duration significantly exceeds that of the transient process) and investigation of the relevant resonant processes.

For the same reason, transient processes interfere into the observation of forced exponential radio pulses and forced exponential video pulses to such an extent that they have prevented from detecting resonance with regard to these signals. That is, contrary to resonance of sustained sinusoidal oscillations, which was detected actually by itself, resonance of exponential radio pulses and resonance of exponential video pulses require some additional efforts to be revealed.

This is quite natural, because nature sometimes prompts scientists to their scientific discoveries (this is how, for instance, radioactivity and penicillin were discovered), however, quite seldom. Usually scientists have to indulge into long and purposeful research to make their scientific discoveries. In our case, investigating resonance at complex

frequencies, we also have to take special measures to minimize the obstructive influence of transient processes on experimental observations.

With all these circumstances in mind, let us get to the discussion of a series of experiments, the results of which:

prove the physical reality of resonance at complex frequencies;

are inexplicable in terms of the current electric circuit theory at real frequencies.

In the first experiment (Antonov, 2008), similar research is performed on an LC-two-terminal (figure 2a) and RL-two-terminal (figure 2b).

Under sinusoidal impact INPU on the LC-two-

terminal circuit, fully in accordance with its frequency characteristic

20

2

iLCi

1Li),0(Z

(7)

the polarity of the output voltage forced component

OUTforcU reverses if the input voltage complex

frequency ip changes. As can be seen, at

0 and 0

the output voltage forced

component is opposite in sign, and at 0 it is

zero.

Quite similarly, under exponential impact INPU on

the RL-two-terminal, fully in accordance with its frequency characteristic

)(LLR)0,(Z 0 (8)

the polarity of the output voltage forced component

OUTforcU reverses if the input voltage complex

frequency p changes. As can be seen, at

complex frequencies of the input exponential voltage

0 and 0 the output voltage forced

component OUTforcU is opposite in sign, and if

0 it equals zero.

Consequently, under exponential influence on the RL-circuit under investigation, processes in it are completely similar to the resonance processes in the electric LC-circuit under sinusoidal impact.

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45

Fig. 2. Signalograms in LC- and RL-two-terminals during investigation of resonance at complex frequencies

The latter result seems inexplicable in terms of the electric circuit theory at real frequencies, because in the aperiodic RL-circuit under investigation any spectral component of the input exponential pulse is known to have a phase shift

not exceeding 2 . It is unclear how these

spectral components managed to form an exponential video pulse with a phase shift.

Thus, naturally, no one ever tried to obtain such a result.

In the second experiment (Antonov, 2008), contrary to the previous one, signalograms in the LC- (figure 3a), RL- (figure 3b) and LCR- (figure 3c) two-terminal circuits are analyzed only at complex resonance frequencies. Therefore, this experiment will allow

extending our knowledge of resonant processes in these electric circuits.

As can be seen (figure 3a), in the LC-two-terminal, under the impact of input sinusoidal

oscillations INPU , the forced components of

voltage drops at the capacitor CforcU and the

inductance coil LforcU are equal in magnitude but

opposite in sign. As a result, the forced component of voltage drop at the oscillatory LC-

circuit CforcLforcOUTforc UUU turns out to be

zero. Therefore, the oscilloscope screen will demonstrate only the relatively short radio pulse

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Fig. 3. Signalograms in LC-, RL- and LCR-two-terminals for resonance at complex frequencies

of the free component of response OUTfreeU . This

is a well-known result.

In the RL-two-terminal circuit (figure 3b), quite similarly, under the influence of input exponential

video pulses INPU the forced components of

voltage drop at the inductance coil LforcU and

the resistor RforcU in the form of exponential

pulses are equal in magnitude but opposite in sign. Therefore, the forced component of the resultant voltage drop in the electric RL-circuit

RforcLforcOUTforc UUU turns out to be zero.

For the same reason, the oscilloscope screen,

once again, will demonstrate only a relatively short pulse of the free component of response

OUTfreeU .

However, two resonant processes in the electric LC- and RL-circuits discussed above are particular cases of a more general resonant process in the electric LCR-circuit, which will be discussed below. As can be seen (figure 3c), similar to the previous resonant processes, in the LCR-two-terminal under consideration influenced

by input damped sinusoidal oscillations INPU at

complex resonance frequency, the forced components of voltage drop at the capacitor

CforcU and the inductance coil LforcU are equal

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in magnitude and phase shifted to each other by a value slightly different from , and voltage drop

at the resistor RforcU is significantly smaller.

Therefore, the forced component of the resultant voltage drop at the oscillatory LCR-circuit

RforcCforcLforcOUTforc UUUU turns to be

zero. Thus, only a relatively short radio pulse of

the free component of response OUTfreeU will be

observed on an oscilloscope screen. This is also an unknown result. This result seems even more inexplicable because there is another well-known and seemingly contradictory result – in an LCR-circuit influenced by sustained sinusoidal oscillations at an allegedly real resonance frequency, it is impossible to get

0UUUU RforcCforcLforcOUTforc .

As can be seen, in all the cases discussed above, the forced component of the output voltage at the LC-, RL- and LCR-two-terminals under

consideration OUTforcU takes zero value at non-

zero values of the forced component of voltage at particular circuit elements of these two-terminals.

Therefore, we must conclude that in all electric circuits under investigation resonance in the true sense of the word is observed; as a result, output voltages at these two-terminals have only the free

component of response OUTfreeU .

Another experiment (Antonov, 2009) not only proves the physical reality of resonance at complex frequencies; it demonstrates how it can be used in radio electronics and communication engineering.

At last the patent (Antonov and Bazhev, 1974) suggests example of practical use of resonance at complex frequencies, which can also be considered as an experiment proving the existence of resonance particularly at complex frequencies.

Since physics is an integral science, the principle of physical reality of imaginary and complex numbers proved above both theoretically and experimentally is a general physical principle, i.e., it is true not only for the linear electric circuit theory (Antonov, 1987; 2008; 2009; 2010c), but for the special theory of relativity (Antonov, 2013b,c), quantum mechanics (Kreidik and Spenkov, 2005; Spenkov, 2014), optics and all other branches of physics, as well.

THE REAL MULTIVERSE

Hidden dimensions of the Universe: However, convincing evidence of the physical reality of imaginary and complex numbers is obviously not enough. Their physical meaning has to be explained, as well. For instance, the existence of a ball lightning is beyond doubt; nevertheless, its physics still remains unknown. Therefore, this physical phenomenon did not have any influence on the development of physics.

In order to understand the physical meaning of imaginary and complex numbers, let us resort to the Euler formula

xsinixcoseix (9)

which can be rewritten in the form

)tsinit(coseett)i(

(10)

As can be seen, in the exponent of the left-hand side of the Euler formula, there is a product of two

physically real values: time t and complex frequency

ip . Therefore. The left-hand side of the

Euler formula is a physically real quantity. Consequently, the right-hand side formula is a physically real quantity, too, including not only the

real component tcoset

, but the imaginary

component, but the imaginary component

tsiniet , as well.

What is the meaning of the physically real imaginary

component tsiniet ? Obviously, it is a

number, and, thus, measures something. Since we cannot either see or touch it, or somehow feel it, it makes sense to refer to it as a hidden dimension (Antonov, 2011a,b,c,d), similar to other hidden dimensions described in (Randall, 2005). Nevertheless, the answer given above is still incomplete, because it does not explain the physical reality behind the hidden dimensions. In other words, what information about the existence of something yet unknown do they suggest? Then, it makes sense to assume that if this unknown exists not in our Universe, it is in some other parallel Universe, and these parallel Universes are somehow connected in a way we do not know.

Parallel Universes: Since the dogma on imaginary numbers having no physical meaning was refuted, the STR has to be adjusted and explained

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accordingly. In this respect, it is first of all noteworthy that the ability of physical bodies to move at superluminal speed does not mean they are directly capable of breaking the light speed barrier, because physical bodies with non-zero rest mass require infinitely large energy to do this.

In view of the above, the STR can be adjusted as follows (Antonov, 2011e; 2012a,b,c; 1013a). In physics, elementary particles moving at subluminal speed are usually referred to as tardyons (or bradyons), and those moving at superluminal speeds – as tachyons. Therefore, we can assume that elementary particles moving at subluminal speed,

whose mass, time and other parameters, in accordance with the relativistic formulae of the STR, are real quantities, are in our tardyon Universe; and elementary particles moving at superluminal speed, whose mass, time and other parameters, according to the relativistic formulae of the STR, are imaginary quantities, for some reasons (see below) are in the tachyon (or anti-tachyon) Universe. These Universes are referred to as parallel for the reason that, despite their boundlessness, they never intersect; although, as shown below, being in the fourth spatial dimension, they are sometimes locally contiguous to each other.

Fig. 4. Graphs of relativistic mass and time, which correspond to formulae (11), (12)

At that, in accordance with the similarity principle of Universes, they are governed by the same physical, chemical and biological laws, however, possibly, in a slightly different way. Therefore atoms and molecules, as well as biological systems in these parallel Universes have the same structure. Only under these conditions it is possible for the inhabitants of other parallel Universes (including rational beings) to visit our Universe. These visits took place in the past and still take place in the present. This means that it is possible for humans to visit other parallel Universes, as well.

In this case, however, the STR relativistic formulae turn out to be incorrect in terms of the tachyon Universe. Indeed, graphs of the relativistic formula (see figure 4a), for instance,

2

0

)cv(1

mm

(11)

where 0m is rest mass;

m is relativistic mass of a moving body;

v is the velocity of the body (e.g., a neutrino);

c is the velocity of light;

have different form at subluminal and at superluminal speeds. Consequently, the formula (11) and others do not correspond to the similarity principle.

Thus they must be adjusted as follows (Antonov, 2013b):

2

0k

2

0k

)c

w(1

m)i(

)kc

v(1

m)i(m

(12)

where cvk is the discrete floor function of

argument cv ;

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kcvw is the local, for each Universe,

velocity which can take values in the range

сw0 ;

v is the velocity measured from our tardyon

Universe, which, hence, can be referred as the tardyon velocity.

As can be seen, at 0k the formula (12)

correspond to the tardyon Universe, and at 1k –

to the tachyon Universe. Its graphs (Figs. 4b) already

look alike. At 2k and at 3k its formula turn out

to correspond to two more parallel Universes which can be referred to as the tardyon Antiverse and the tachyon Antiverse, or the anti-tardyon Universe and

the anti-tachyon Universe. Then, at 4k , as can

be seen, we get once again formula (12) for the tardyon Universe.

Above, we have proven the principle of physical reality of not only imaginary, but complex numbers, as well; due to this, any intermediate parallel Universes can exist: tardyon-tachyon, tachyon-anti-tardyon, and others, which are not shown in figures 4b for simplicity.

This set of parallel Universes is referred to as the Multiverse.

The structure of the Multiverse: At present, quite a few hypotheses of various Multiverses have been suggested. Among the most recent ones, (Lewis, 1986; Linde, 1990; Greene, 1999; Deutsch, 2002; Tegmark, 2003; Ellis et al, 2004; Steinhardt and Turok, 2007; Carr, 2009; Lucash and Mikheyeva, 2010; Greene, 2011) can be mentioned. However, their common fundamental shortcoming is the fact that they cannot be experimentally verified either now or in a distant future (Conley et al., 2007; Ellis, 2011).

The hypothesis of the Multiverse structure discussed herein (Antonov, 2011e; 2012a,b,c; 2013a) is very likely to turn into a theory in the nearest future, because it can be convincingly verified by people visiting other parallel Universes through portals. To make sure this is really so, it is necessary just to organize expeditions equipped accordingly, and to enter these parallel Universes through natural portals, which correspond to points of contact (and certain mutual penetrations) of adjacent parallel Universes. This will be one more experiment refuting the dogma of the STR.

To this end, it is necessary to know, at least approximately, the ways of detecting these natural

portals, the reasons of portal formation between adjacent parallel Universes, the portal formation mechanism, the necessary portal navigation means, the structure of the Multiverse, the differences between parallel Universes, the dangers for people visiting parallel Universes, and many other things.

Let us try to answer some of the questions to the extent it is possible with scarce initial experimental data. First, let us say a few words about the structure of the Multiverse. It was mentioned above that at

0k and at 4k the formula (12) correspond to

the tardyon Universes. However, these can be different tardyon Universes. Then the Multiverse will have the spiral structure.

If we assume that at 0k and at 4k the

formula (12) correspond to one and the same tardyon Universe (ours), the Multiverse will have the ringed structure.

It is possible to assume that different configurations of spiral-ringed Multiverses can exist, as well.

Further research will show which of the structures is correct.

One way or another, in all Multiverse structures the closest to our tardyon Universe are the tachyon Universe and the tachyon Antiverse, which separate it from the tardyon Antiverse. Similarly, the tachyon Universe is always separated from the tachyon Antiverse by the tardyon universe and the tardyon Antiverse. Thus, annihilation of both the tardyon Universe and the tardyon Antiverse, as well as annihilation of the tachyon Universe and tachyon Antiverse, is ruled out.

Furthermore, it is easy to note that in these different parallel Universes, as follows from the formula (12), for an external observer time flows in different directions – in the reverse direction in the tardyon Universe, and in perpendicular direction with regard to our Universe in the tachyon Universe and in the tachyon Antiverse. It is not ruled out that the velocity of time is also different. However, for the inhabitants of all parallel Universes, their own time flows in only one direction – from the past into the future.

Portals: It is also easy to see that, in accordance with the formula (12) and other relativistic formulae modified this way, tardyon velocities v in different

parallel Universes differ by the value which is multiple of the velocity of light c . On the other hand, as noted

above, physical bodies with a non-zero rest mass

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0m are unable to cross the light barrier and, thus,

make a transition from one parallel Universe into another. Then how do elementary particles get from the tardyon Universe into the tachyon Universe, and vice versa?

To clarify the issue, let us once again resort to the Euler formula, this time, rewriting it in the form

tsinitcoseti (13)

It can be seen that the real component tcos in our

tardyon Universe is inseparably linked to the

component tsini in the tachyon Universe, which

is perceived as a hidden dimension in the tardyon Universe. However, if the Euler formula is rewritten in the form

tcositssine)2t(i

(14)

it can be interpreted as follows. Earthlings, having got into the adjacent tachyon Universe, will perceive it

already as the tardyon Universe. For them, the real

component tsin in will correspond to the real

dimensions, and the component tcosi ‒ to the

hidden extra dimensions.

In other words, to make a transition from one parallel Universe into another, it is necessary to shift the

oscillation phase at frequency by 2 . That is, in

portals, which are the transition zone between adjacent parallel Universes, this phase must

gradually shift by 2 or by 2 . This

transition is similar to the gradual transition from air into water and backwards when sea bathing.

Now we just have to understand the phase of which

oscillations must be gradually shifted by 2 in

portals. Since natural portals are usually large territorial formations which open and close quite slowly, these oscillations are, obviously, infra-low-frequency oscillations. Presumably, they are caused by the rotation of the Earth around its axis and/or revolution of the Moon about the Earth. Most likely, the nature of these oscillations is electromagnetic. This assumption is likely to be confirmed by the experiment with the American destroyer Eldridge.

What kind of natural phenomena can determine the appearance and disappearance of natural portals? This question definitely deserves special consideration. It can be assumed that it is determined

by the geotectonic processes causing shock oscillations in the Earth crust, which call forth the phase shift of the resultant infra-low-frequency oscillations.

Finally, let us give a comparison explaining why transition from one parallel Universe into another through portals, based on the Euler formula, is possible; however, it is impossible based on the Lorentz-Einstein formula. This is similar to the way it is possible to go from one room of your house into another through the door, but it is impossible through the wall separating them.

In the Multiverse, parallel Universes are constantly exchanging elementary particles and living beings through numerous portals. This is why tachyons exist. This is why unknown animals and humanlike creatures, referred to as snowmen, yeti, or bigfoots, sometimes appear in our tardyon Universe; they may know the locations of the most convenient and safe portals and use them.

CONCLUSION

Parallel Universes discussed above obviously have so much in common, in accordance with the similarity principle, that they are compatible. Therefore, in particular, mutual visits of inhabitants of different parallel Universes are possible through portals.

However, let us recollect that time in them, presumably, flows in different directions. For this reason, it is possible to return into a different time period after visiting other parallel Universes.

Other differences and hazards are also possible. For instance, astronomic maps in different parallel Universes are most likely to be significantly different. Therefore, deep space travel is most advisable to be made through the labyrinths of portals and different parallel Universes. This may allow significantly (even by several orders of magnitude) reducing the duration of these journeys.

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