SPECIAL THEORY OF RELATIVITY PRESENTED IN PHYSICS TEXTBOOKS IS WRONG Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

PHYSICAL SCIENCES

SPECIAL THEORY OF RELATIVITY PRESENTED IN PHYSICS TEXTBOOKS IS WRONG 1

Antonov A.

Ph.D., HonDSc, H.ProfSci, ResProf Independent Researcher, Kiev, Ukraine

Abstract

It is shown that the entire practice of using Ohm's law in the theory of linear AC circuits is a theoretical and experimental evidence of physical reality of imaginary numbers. And therefore, the version of the special theory of relativity (SRT) set forth in all physics textbooks, denying physical reality of imaginary numbers through the use of the principle of light speed non-exceedance, is incorrect.

Keywords: imaginary numbers; Ohm's law; resonance; transient processes; special theory of relativity.

1. Introduction

The currently generally accepted version of the special theory of relativity (SRT), created at the beginning of the 20th century by Joseph Larmor, Nobel laureate Hendrik Antun Lorentz, Jules Henri Poincaré, Nobel laureate Albert Einstein and other prominent scientists, which is now studied in all university and even school physics textbooks, is incorrect [1], since from the postulated principle of not exceeding the speed of light in it follows the statement about the physical unreality of imaginary numbers, which was refuted experimentally. Consequently, statements about physical unreality of imaginary numbers and uniqueness of our visible universe following from this principle also turned out to be incorrect.

Besides, this postulate has been relevant for the SRT only because relativistic formulas obtained in the theory at superluminal velocities correspond to imaginary mass, imaginary time and other imaginary physical quantities, the meaning of which could never be explained by anyone.

In this regard, it is appropriate to recall that imaginary numbers appeared in mathematics about 500 years ago in the works of Scipione del Ferro, Niccolo Fontana Tartaglia, Gerolamo Cardano, Lodovico Ferrari and Rafael Bombelli [2]. And [3] even asserts that imaginary numbers were discovered even before by Paolo Valmes, who was sentenced to death at the stake by Spanish inquisitor Tomás de Torquemada for this discovery.

However, despite the fact that in subsequent years a perfect theory of functions of a complex variable was created by efforts of outstanding mathematicians Abraham de Moivre, Leonhard Euler, Jean Le Rond D'Alembert, Caspar Wessel, Pierre-Simon de Laplace, Jean-Robert Argand, Johann Carl Friedrich Gauss, Augustin Louis Cauchy, Karl Theodor Wilhelm Weier-strass, William Rowan Hamilton, Pierre Alphonse Laurent, Georg Friedrich Bernhard Riemann, Oliver Heav-iside, Jan Mikusinski and others, it did not explain physical sense of imaginary numbers.

Although imaginary numbers are now widely used in all exact sciences, such as optics, electrical engineering, radio electronics, mechanics, hydraulics, acoustics, etc, they neither explain physical sense of imaginary numbers. However, in contrast to the existing version of the SRT, these sciences have never denied physical sense of imaginary numbers over the past 500 years, even not knowing how to explain it.

It is not surprising that not everyone among physicists agreed with such a simple solution to the complex problem of explaining physical sense of imaginary numbers proposed by the creators of the SRT. Therefore, such complex experiments as MINOS carried out at the American Tevatron Collider and OPERA carried out at the European Large Hadron Collider took place. They aimed at refuting the postulated principle of light speed non-exceedance by detecting neutrinos moving with superluminal velocities, and thereby proving physical reality of imaginary numbers. However, physical community found the results of these experiments unreliable and ignored them.

Nevertheless, other experiments [4] - [8] were carried out at the same time. They actually solved the problem. That is, they refuted the principle of light speed non-exceedance. These experiments can be repeated and verified in any radio electronic laboratory and, therefore, are absolutely reliable and evidential. The existing version of the SRT actually collapsed like a house of cards without the principle of light speed non-exceedance that had been refuted.

Therefore, let's give a brief description of these and subsequent [9] - [15] very important and successful radio engineering experiments, alternative to the unsuccessful MINOS and OPERA physics experiments.

2. Proof of physical reality of imaginary numbers using Ohm's law in the interpretation of Steinmetz

Ohm's law in the interpretation of Steinmetz [16] is used literally at every turn when studying any linear electrical circuits. This well-known law appears to prove physical reality of concrete2 imaginary numbers

1 This is an extended version of the article "Antonov A.A. The special theory of relativity taught in all physics textbooks is wrong. International Conference of Eurasian Scientific Association. Theoretical and practical issues of modern science. Moscow: ESA. 2021. 7(77). 11-15. (In Russian) www.esa-conference.ru"

2 That is, provided with references to physical units, such as meter, kilogram, volt, etc.

[17]-[21] in addition to its possibility to be used for engineering calculations. Let's make sure of this.

Only three different types of passive electrical elements are used in the theory of linear electric circuits. They are resistors R , capacitors C and inductors L. Dependence between the electric current i(t)

flowing through them and the voltage u(t) applied to them is generally described as follows

i(, )-u(% „(, )-Ldi(%t i(t ) -

(1a) (1b) (1c)

Algebraic dependence (1a) is called Ohm's law for direct current circuits. And the dependences (1b) and

(1c) between the functions i(t) and u(t) are apparently differential-integral and strictly do not correspond to Ohm's law. But in one special case important for

practice, for sinusoidal currents i(t ) and voltages

u(t) , formulas (1b) and (1c) can also be converted to algebraic expressions

u(t) = jaLi(t) (2a)

i(t) = jaCu(t) (2b)

using the symbolic method proposed by Charles Proteus Steinmetz, according to which the real physical effect is re-placed by the effect corresponding to the

Euler formula. Imaginary unit is designated here as j ,

since designation I in the theory of electrical circuits is used for electric current.

And the algebraic expressions (2a) and (2b) correspond to Ohm's law, as can be seen. Moreover, the

value Xl = jaL is the electrical resistance of an inductor, and the value Xc = 1 jaC = — j/aC is the electrical resistance of a capacitor. These electrical resistances are measured by imaginary numbers opposite in sign, in contrast to the electrical resistance of a resistor, measured by real numbers.

Therefore, the magnitude of complex resistance of any alternating current circuit containing not only resistors R , but also capacitors C and/or inductors L depends on frequency. Thus, in accordance with Ohm's law in the interpretation of Steinmetz the magnitude of flowing current depends on frequency. Moreover, this dependence is easily determined using devices available in any radio electronic laboratory.

And if the postulate of physical unreality of imaginary numbers were correct, then the magnitude of flowing current in any alternating current circuit would have to depend only on the magnitude of resistance and never on the magnitude of capacitance and inductance. Therefore, current magnitude should not change when oscillation frequency of sinusoidal voltage applied to such an electric circuit changes. But any experiment disproves this assumption.

Figure. 1. This is all that is needed instead of the Large Hadron Collider for the experimental proof ofphysical

reality of imaginary numbers.

Thus, evidence of physical reality of imaginary numbers that could not be obtained by the MINOS experiment at the Tevatron collider and the OPERA experiment at the Large Hadron Collider turns out to be

de facto obtained3 long ago by millions of engineers all over the world and it is daily confirmed by their practical activities in all radio electronic laboratories.

3 Consequently, the generally accepted version of the SRT could have been refuted back in the 19th century, i.e. before its creation, if someone needed it at that time

Notably, this plain evidence of the principle of physical reality of imaginary numbers obtained by measuring required parameters of electric circuits with devices, such as testers (see Figure 1), oscilloscopes, etc., available in any radio engineering laboratory is the most convincing.

After all, it is exactly the ability to register with devices X-ray, radioactive, ultraviolet and infrared radiation, infra and ultrasound, magnetic field, atoms and subatomic particles, as well as many other physical entities that are not registered by the human senses, proves their physical reality. Why, then, proving physical reality of imaginary numbers required the unique expensive OPERA and ICARUS experiments at the Large Hadron Collider involving several hundred professors instead of a simple and cheap experiment using an ordinary tester in physics and involving only one radio engineer? That is because physical community did not actually need the proof and scientific truth. They only needed to create impression that refutation of the principle of light speed non-exceedance in the SRT was an extremely difficult matter, due to which the principle was actually irrefutable, and the version of the SRT presented in textbooks was correct.

In fact, since mathematics is the single universal language of all exact sciences, correct mathematical interpretation of radio engineering and any other experiments is indisputably convincing for all other exact sciences. After all, the Nature is integral. And only people created many sciences to describe it due to their limited intellectual capacity. However, it is natural that sciences cannot refute each other.

Thus, since the principle of physical reality of imaginary numbers in the SRT has been experimentally proved, there is no need for the postulated principle of light speed non-exceedance and for corrected relativ-istic formulas that allow explaining the SRT at superlu-minal velocities

3. Proof of physical reality of imaginary numbers as a result of study of resonance

Resonance was discovered by Galileo di Vincento Bonaiuti de'Galilei back in 1602 [22]. But all textbooks give only a description of near-resonance processes at real frequencies, whereas a perfect theory of resonance at complex frequencies [4]-[7], [9], [11], [14] has not yet been provided in any textbook of physics.

Indeed, attributes of resonance in electric LC-circuits are considered to be:

• extreme value of the forced component of response at resonance frequency;

• zero phase shift between the impact and the forced component of response at resonance frequency;

• equality of resonance frequency and frequency of free (in particular, shock) oscillations.

However, in the simplest electric LCR-circuits these attributes manifest them-selves only approximately. Therefore, in accordance with its current incorrect (or ra-ther, approximate) interpretation in most of oscillation LCR-circuits:

• different real resonance frequencies correspond to the first and the second at-tributes of resonance mentioned above;

• several (two for second-order circuits) real resonance frequencies usually correspond to each of the above-mentioned attributes of resonance;

• resonance frequencies never equal to frequency of free oscillations, as shown by Leonid Isaa-kovich Mandelstam [23].

Actually, difference of resonance frequency and frequency of free oscillations from the frequency is insignificant and does not usually exceed experimental error. However, difference between the speed of light and the speed of neutrino in the MINOS and OPERA experiments4 was also insignificant and did not exceed experimental error. Nevertheless, some dozens of articles analyzing possible experimental errors were published following several months after publication of the OPERA experiment results. And their result was the ICARUS experiment claiming that the OPERA experiment was wrong.

At the same time, although the results of theoretical and experimental studies of resonance in electric LCR-circuits, proving physical reality of imaginary numbers, were published more than ten years ago, they still have been neither refuted nor commented on. Even despite the fact that the principle of physical reality of imaginary numbers in these studies is confirmed by existence of television and telecommunications, radiolocation and GPS navigation, resonance, Ohm's law and so on.

4. Proof of physical reality of imaginary numbers as a result of study of transient processes

Modern algebra is self-contradictory. For example, to solve quadratic equations two algorithms are usually applied, of which one using well-known real numbers and the other using incomprehensible complex numbers. Moreover, in the first case it is argued that sometimes there can be two solutions, sometimes one solution, and some times no solution at all.

In the second case it is argued that there are always two solutions, which can be both real and complex numbers. At the same time, the obvious fact that two different mutually exclusive solutions (when 2

b — 4ac < 0 ) cannot exist is ignored.

So where is the truth? Which solution is true? In the formal logic, the Latin aphorism 'Tertium non da-tur', i.e, there is no in-between, corresponds to this situation.

Since a purely mathematical convincing answer to the question posed has not yet been found, we shall use a physics experiment [8], [9], [14]. Let us consider the so-called transient processes that for whatever reason (usually under external influence) correspond to transition of any energy-intensive system from one energy state to another. They are due to the fact that energy change can never be naturally instantaneous. Therefore, such a transient process always takes some time.

And determination of transient process parameters requires solving of the so-called algebraic characteristic

4 The purpose of which, just like of this study, was to prove physical reality of imaginary numbers.

equations that are in a certain way connected with differential equations describing behavior of such systems.

Transient processes are met everywhere. For example, they include oscillations of a pendulum after being pushed by someone. Therefore, everyone can perform this simple physics experiment and make sure that:

• oscillatory transients exist;

• oscillations are damped;

• damping of oscillations takes some time.

These experimental data are quite enough in order

to get an answer to the above-posed question of which of the solutions of the algebraic equation is correct.

An aperiodic transient process turns out to correspond to solution of the algebraic characteristic equation ax2 + bx + c = 0 at a positive value of the dis-2

criminant b — 4ac > 0 . A critical transient process corresponds to solution of the quadratic characteristic

equation at b — 4ac = 0. And oscillatory transient process (in the form of damped oscillations; let's recall here the movement of a pendulum after a push) corresponds to solution of the quadratic characteristic equa-2

tion at b — 4ac < 0 .

They all really exist. Moreover, for the cases when

2 2 b — 4ac > 0 and b — 4ac = 0 , there are coinciding solutions of algebraic quadratic equations according to both of the above algorithms using real and complex numbers.

2

However, when b — 4ac < 0, there is a solution in the form of complex numbers. So, there is oscillatory transient process corresponding to such a solution. When b — 4ac < 0 , there is no solution in the form of real numbers. And therefore, there should be no transition process corresponding to such a solution. Nevertheless, everyone can make sure that oscillatory transient process exists just pushing a pendulum.

Similar conclusions can be drawn for solutions of higher order algebraic characteristic equations. And such conclusions are experimentally confirmed by bell ringing and piano music, tsunami and swinging children's swings, Indian summer (or, conversely, spring frosts) and many other natural and man-made phenomena.

Therefore, it is to be concluded that the only true solution to algebraic equations is a solution in the form of complex numbers. In that case, complex (and, consequently, imaginary) numbers should be recognized as physically real.

5. The SRT studied in all physics textbooks is incorrect

Since the Nature is integral and non-contradictory, the Science trying to explain it must also be integral and non-contradictory. Therefore, it is unacceptable that different scientific theories are inconsistent. Consequently, the principle of physical reality of imaginary numbers proven in the theory of electrical circuits must be recognized as general scientific and all scientific theories must be corrected taking into account the principle.

Nevertheless, despite even several of the above-mentioned experimental proofs of the principle of physical reality of imaginary numbers [4]-[15], [17]-[21], the principle of light speed non-exceedance is still believed to be true in all physics textbooks. And textbooks still set out the generally accepted, but incorrect version of the SRT.

Thus, as shown above, stating that imaginary numbers are physically unreal, authors of these textbooks actually deny physical realities that undoubtedly exist next to them and are known to everyone, in particular, television and telecommunications, radiolocation and GPS navigation, bell ringing and piano music, tsunami and 'Indian summer', children's swing, resonance, Ohm's law and so on. Authors of these textbooks have neither understood nor explained to readers that all this refutes the SRT. This circumstance naturally raises doubts about the accuracy of presentation of other sections of such textbooks. For example, sections related to astrophysics explaining physical processes and objects that are very distant from us, much more distant and difficult to understand than bell ringing and children's swings.

6. Conclusions

However, It is regrettable that authors of physics textbooks, as well as scientific literature setting out and popularizing modern physics are unaware of the fact that fundamental scientific laws (Ohm's law and resonance) and well-known natural phenomena and man-made processes (tsunami, sound of church bells, piano music, swinging children's swings, television, radiolocation, etc.) indisputably prove physical reality of imaginary numbers5 . Therefore, currently existing versions of the theory of relativity, quantum mechanics and other exact sciences need to be corrected accordingly.

In this regard, scientific works of the 1988 Nobel Prize laureate Melvin Schwartz [24], who proved that development of the theory of relativity is impossible without a deep understanding of problems of both electrodynamics and electrical engineering, as well as works of the 1965 Nobel Prize laureate Julian Seymour Schwinger [25], who proved that development of quantum mechanics is impossible without a deep understanding and solution of problems of electrodynamics are of out-standing significance.

The generally accepted version of the special theory of relativity presented now in physics textbooks is incorrect, since all the relativistic formulas obtained in the theory are incorrect, they have been incorrectly explained using the incorrect principle of light speed non-exceedance and entailed wrong conclusions consisting in existence of only our visible universe and physical unreality of imaginary numbers [26]-[29].

The corrected alternative version of the SRT is proposed in [29]-[32].

Acknowledgements

The author is grateful for participation in the discussion of the paper to Olga Ilyinichna Antonova, whose criticism and valuable comments contributed to improvement of the paper.

5 In the Thirty Years' War Cardinal Richelieu, prompted by similar considerations, ordered to inscribe upon cannons the

following text: "Ultima ratio regum". And the last argument of scientists is experiments.

REFERENCES:

1. Antonov A.A. (2014). Verification of the second postulate of the special relativity theory. Global Journal of Science Frontier Research A: Physics and Space Science. 14(3). 51 - 59.

2. Weisstein E.W. (2005). The CRC Concise Enciclopedia of Mathematics. 3-rd ed. CRS Press. Roca Raton, FL

3. Beckmann P. (1976). A History of n. 3-rd edition. St. Martin'sPress. NY.

4. Antonov A.A. (2008). Physical Reality of Resonance on Complex Fre-quencies. European Journal of Scientific Research. 21(4), 627 - 641

5. Antonov A.A. (2009) Resonance on Real and Complex Frequencies. Eu-ropean Journal of Scientific Research. 28(2), 193 - 204

6. Antonov A.A. (2010) New Interpretation of Resonance. International Journal of Pure and Applied Sciences and Technology, 1(2), 1 - 12.

http://doi.org/10.17686/sced_rusnauka_ 2010-888

7. Antonov A.A. (2010) Oscillation processes as a tool of physics cognition. American Journal of Scientific and Industrial Research, 1(2), 342 - 349.

doi:10.5251/ajsir.2010.1.2.342.349

8. Antonov A.A. (2010) Solution of algebraic quadratic equations taking into account transitional processes in oscillation systems. General Mathematics Notes, 1(2), 11 - 16. http://doi.org/10.17686/sced_rusnauka_2010-887

9. Antonov A. A. (2013) Physical Reality of Complex Numbers.. International Journal of Management, IT and Engineering. 3(4). 219 - 230.

http://doi.org/10.17686/sced_rusnauka_2013-898

10. Antonov A.A. (2014) Correction of the special theory of relativity: physical reality and nature of imaginary and complex numbers. American Journal of Scientific and Industrial Research. 5(2) 40-52. doi:10.5251/ajsir.2014.5.2.40.52

11. Antonov A.A. (2015) Physical reality of complex numbers is proved by research of resonance. General Mathematics Notes. 31(2) 34-53

http://www.emis.de/jour-nals/GMN/yahoo_site_admin/assets/docs/4_GMN-9212-V31N2.1293701. pdf

12. Antonov A.A. (2015) The principle of the physical reality of imaginary and complex numbers in modern cosmology: the nature of dark matter and dark energy. Journal of Russian physical and chemical society. 87(1) 328-355 (in Russian)

13. Antonov A.A. (2016) Physical Reality and Nature of Imaginary, Complex and Hypercomplex Numbers. General Mathematics Notes. 35(2) 40-63

14. Antonov A.A. (2017) The physical reality and essence of imaginary numbers. Norwegian Journal of development of the International Science. 6. 50-63

http ://www. nj d-iscience. com

15. Antonov A. A. (2018) Physical Reality and Essence of Imaginary Num-bers in Astrophysics: Dark Matter, Dark Energy, Dark Space. Natural Science. 10(1), 11-30.

16. Steinmetz C.P., Berg E.J. (1900). Theory and Calculation of Alternating Current Phenomena. Electrical World and Engineer Inc. NY.

17. Antonov A.A. (2015) Ohm's law explains astrophysical phenomenon of dark matter and dark energy. Global Journal of Physics 2(2) 145-149

http://gpcpublishing.com/index.php?jour-nal=gjp&page=arti-

cle&op=view&path%5B%5D=294&path%5B%5D=p df_14

18. Antonov A.A. (2015). Adjustment of the special theory of relativity ac-cording to the Ohm's law. American Journal of Electrical and Electronics Engi-neeing. 3(5) 124-129. doi: 10.12691/ajeee-3-5-3

19. Antonov A. A. 2016. Ohm's Law Refutes Current Version of the Special Theory of Relativity. Journal of Modern Physics, 7(16), 2299-2313.

DOI: 10.4236/jmp.2016.71619820.

20. Antonov A. A. (2016) Ohm's Law explains phenomenon of dark matter and dark energy. International Review of Physics. 10(2). 31-3

https://www.praiseworthyprize.org/jsm/in-dex.php?journal=irephy&page=article&op =view&path%5B%5D= 18615

21. Antonov A. A. (2016) Ohm's Law is the general law of exact sciences. PONTE. 72(7) 131-142. doi: 10.21506/j.ponte.2016.7/9

22. Frova A., Marenzana M. (2006). Thus spoke Galileo: The great scientist's ideas and their relevance to the present day. Oxford University Press. NY. 133137.

23. Mandelshtam L.I. (1955). Lectures on Theory of Oscillations. vol. 4. Moscow: Publishing House of AS of the USSR. (In Russian)

24. Schwartz M. (1987). Principles of Electrodynamics. Dover Publications, NY.

25. Schwinger, J. S. (Ed.). (1958). Selected Papers on Quantum Electrodynamics. Dover Publications, NY.

26. Antonov A. A. (2019) The special theory of relativity in the 20-th century was not and not be created. Journal of Russian physical and chemical society. 91(1). 57-94. (in Russian)

27. Antonov A. A. 2020. Einstein was wrong: according to WMAP and Planck spacecraft research we live in a six-dimensional hypercomplex space. Österreichisches Multiscience Journal (Innsbruck, Austria). 2020. 35(1). 61-72. http://osterr-science.com.

28. Antonov A. A. (2020) Albert Einstein was ahead of his time: he did not complete the existing version of the special theory of relativity due to the lack of ex-perimental data obtained only in the 21st century. Journal of Russian physical and chemical society. 92(1). 39-72. (In Russian)

29. Antonov A. A. (2020) Comparative Analysis of Existing and Alternative Version of the Special Theory of Relativity. Journal of Modern Physics. 11(2), 324-342. DOI: 10.4236/jmp.2020.112020

30. Antonov A. A. (2021) Relativistic formulas of the existing version of the special theory of relativity are incorrect, they are incorrectly explained and incorrect conclusions are drawn from them. German International Journal of Modern Science. 4(1). 38-4730.

31. Antonov A. A. (2016) What Physical World Do We Live in. Journal of Modern Physics, 7(14), 1933-1943 DOI: 10.4236/jmp.2016.714170

32. Antonov A. A. 2021. The physics of imaginary numbers is the new physics, physics of the invisible but really existing world. Norwegian Journal of development of the International Science 54(1). 38-46. http ://www. nj d. iscience.com