This paper is an addendum to the article “On the Evolution of Approaches to the Space- Time Symmetry”, Natural Science, Vol. 10, No. 3, pp. 81-84, 2018. The analysis is based on the Galileo’s postulate that all inertial reference frames are equal. The postulate results immediately in the absolute speed limit, which is identical for all bodies in all reference frames. Two kinds of elementary particles are admitted: unhurried and restless ones (ex-emplified, accordingly, with electrons and photons). Any particle may be treated as wave packet, the particle mass and momentum depending on the reference frame. Thus, space-time-symmetry is the natural background for the relativistic and quantum theories

Petelin, Mikhail

Thumm, Manfred

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https://www.scirp.org/journal/ns Natural Science, 2020, Vol. 12, (No. 9), pp: 661-665   https://doi.org/10.4236/ns.2020.129056 661 Natural Science   On the Elementary Mechanical Effects of the Space-Time-Symmetry Relativity Mikhail Petelin1, Manfred Thumm2 1Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia; 2Institute for Pulsed Power and Microwave Technology, Karlsruhe Institute of Technology, Karlsruhe, Germany Correspondence to: Mikhail Petelin, ; Manfred Thumm,  Keywords: Absolute Speed Limit, Restless and Unhurried Elementary Particles, Particle Wave Packets, Mass-Momentum Relation Received: July 29, 2020      Accepted: September 24, 2020      Published: September 27, 2020 Copyright © 2020 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/     ABSTRACT This paper is an addendum to the article “On the Evolution of Approaches to the Space- Time Symmetry”, Natural Science, Vol. 10, No. 3, pp. 81-84, 2018. The analysis is based on the Galileo’s postulate that all inertial reference frames are equal. The postulate results im-mediately in the absolute speed limit, which is identical for all bodies in all reference frames. Two kinds of elementary particles are admitted: unhurried and restless ones (exemplified, accordingly, with electrons and photons). Any particle may be treated as wave packet, the particle mass and momentum depending on the reference frame. Thus, space-time-sym- metry is the natural background for the relativistic and quantum theories.  1. GALILEO’S REFERENCE FRAMES Our methodical paper [1] was stimulated by the assumption of Refs. [2, 3] that the number of inde-pendent relativistic postulates [4, 5] might be minimized. This economic approach leads to the following simple corollaries.  Figure 1 shows the famous Galileo’s (anti-Aristotle) model of inertial reference frames: two boats moving on a lake [6]. Each of these frames carries its own coordinate-time system ,ξ τ  or ,ξ τ′ ′ . The measurement units being taken identical, the symmetry-reciprocity of the reference frames is met with the following equations [1] ξ γξ κτ ξ γξ κττ κξ γτ τ κξ γτ′ ′ ′= + = −⇔′ ′ ′= + = − +                              (1) being left-right cyclic recurrent due to the interrelation between the constant coefficients γ  and κ  2 2 1γ κ− = .                                     (2) Open Access  https://doi.org/10.4236/ns.2020.129056 662 Natural Science  Under the following Minkowski change of variables [4, 5] ˆ ˆi , iτ τ τ τ′ ′= =                                  (3) the transformation (1) becomes equivalent to the mutual rotation of Cartesian coordinates (Figure 2) [4, 7], and the cyclic recurrence condition (2) turns out to be equivalent to the Pythagoras theorem 2 2sin cos 1α α+ = .                               (2a) According to (1), the relative velocity of the reference frames is β κ γ= .                                    (4) Equation (4) and the Pythagoras condition (2), (2a) give 211γβ=−.                                  (5) The Galileo’s space-time symmetry (1) and (2) results in the elementary kinematic effects [4, 5]: the “Lorentz length contraction” and the “paradox of twins”. 2. FREE BODY KINEMATICS According to (1), (2) [1, 4, 5], if a body in Figure 1 moves with a velocity bβ ′  relative to the ,ξ τ′ ′  reference frame, the body velocity relative to the ,ξ τ  frame is  1bbbβ ββββ′ +=′+.                                  (6)   Figure 1. Galileo’s inertial reference frames [1].   Figure 2. Mutual rotation of Cartesian coor-dinate systems [7] ˆ,ξ τ  and ˆ,ξ τ′ ′ . 'ξξτ 'τ⇒  https://doi.org/10.4236/ns.2020.129056 663 Natural Science  It is natural to postulate that the body mass m and the body momentum (quantity of body motion) p compose a vector [ ]ˆ ,m p  similar to the time-coordinate vector [ ]ˆ,τ ξ  in the Minkowski scheme Figure 2 (as time may be arbitrary at zero coordinate, so mass may be arbitrary at zero momentum). Therefore, by analogy with the time-coordinate transformation (1), the mass-momentum transformation is p p m p p mm p m m p mγ κ γ κκ γ κ γ′ ′ ′= + = −⇔′ ′ ′= + = − +                          (7) The Galileo’s space-time symmetry (1), (2) and (7) admits two kinds of elementary particles: restless and unhurried ones. 2.1. Restless Particles According to the formulas (6) and (7), a particle may move in any reference frame with the ultimate velocity: 1b bβ β ′= =                                    (8) the particle mass and the particle momentum being interrelated with the formulas  ( ) ( ) ( ) ( )m m p p m m p pγ κ γ κ γ κ γ κ′ ′ ′ ′= + = + = ⇔ = − = − =               (9) (so the mass = momentum is non-zero in any reference frame).  2.2. Unhurried Particles If a body is resting in the reference frame ,ξ τ′ ′ : 0 00, 0,b p m mβ ′ ′ ′= = = ,                          (10) then, according to (7), the mass and the momentum of the body in another reference frame ,ξ τ  are 0m mγ=                                   (11) p mβ= .                                  (12) Therefore, as it follows from (11) and (5), if the body velocity bβ  approaches the universal speed limit, the body mass m approaches infinity. For this reason, no real force can accelerate any unhurried particle up to the ultimate velocity equal to 1. 2.3. Common Speed Limitation According to (6)-(12), both restless and unhurried particles undergo the absolute speed limitation , 1b bβ β ′ ≤ ,                                 (13) corresponding to the 12th Theorem of Proclus (proved on the contrary) [8]: “During a limited time it is not possible to go an infinite distance”. 3. WAVE APPROACH At the frame-to-frame transition described by (1), (2), and (7), and illustrated with Figure 2, the sca-lar product of the vectors [ ]ˆ ,m p  and [ ]ˆ,τ ξ  remains invariant: m p m pτ ξ τ ξ′ ′ ′ ′− = − ,                          (14) which may be attributed to a phase of a relevant wave ( )e i m pτ ξ− − .                                 (15) 3.1. Restless Particles If a particle is restless, according to (8) and (9), the wave (15) takes the form   https://doi.org/10.4236/ns.2020.129056 664 Natural Science  ( ) ( ) e eim ipτ ξ τ ξ′ ′ ′− − − −= ,                              (16) where the frequencies-propagators m p= , m p′ ′=  are interrelated with the formulas (9) corresponding to the Doppler effect [9]. The waves (16) may be composed in a packet  ( ) ( )e dimmg mτ ξτ ξ+∞− −−∞Φ − = ∫                            (17) propagating with the ultimate speed equal to 1. 3.2. Unhurried Particle If an unhurried particle, according to (10) - (12), is weakly relativistic (in a relevant reference frame) 1κ  ,                                       (18) the invariant (14) reduces to ( )0 0m p m m κτ ξ γτ κξ τ θ− = − ≈ + ,                        (19) where  20 2mκκθ τ κξ = −  .                              (20) The corresponding waves (15) of a common resting mass 0m  may be composed in a packet  ( ) ( ), e di m pg τ ξκξ τ κ+∞− −−∞Ψ = ∫ ,                            (21) which, with account of (19), may be written in the form  0e im τψ −Ψ ≈ ,                                  (22) where the envelope  ( ), e dkig θκψ ξ τ κ+∞−−∞= ∫                                (23) slowly depends on time and coordinate. By using (20), one finds that ( ),ψ ξ τ  satisfies the following pa-rabolic equation 2202imψ ψτ ξ∂ ∂= −∂ ∂.                                 (24) 4. CHANGE TO PRACTICAL SCALES The Galileo-symmetry-defined variables [used in the relations (1)] are matched to the ordinary space-time variables x, t (used in the Lorentz transformation [4, 5]) with the substitution [1] x xct ctξ ξτ τ′ ′= =′= =                                (25) where c is the ultimate body velocity that should be measured in the usual units. Relevant experiments were performed primarily with the light [9], which is an electromagnetic wave (16) composed of restless photons. The photon dimensional speed was measured to be 83 10 m sc = ×  [9]. The wave (16) may be written in the form ( ) e i t kxω− − , where ω  is the angular frequency and k cω=  is the wave vector. In accordance to [5, 9], the “usual” photon energy is ω  and the “usual” photon mo-mentum is k , where   is the Planck constant divided by 2π.   https://doi.org/10.4236/ns.2020.129056 665 Natural Science  Results of subsequent experiments with unhurried charged particles (e.g. electrons) [10] proved to agree with the Galileo’s space-time symmetry as well. The mass-momentum relations (11), (12) corres-pond with the famous Einstein’s formula 2E Mc=  [4, 5, 11] (Einstein in his original paper [11] uses L for the particle energy E and V for the velocity of light c). At the coordinate-time change (25), if the “symme-tric” resting mass 0m  and the “practical” mass em  of the electron are linearly interrelated 0 em m c= , the electron wave-packet equation (24) may be converted to a form corresponding to the Schrödinger eq-uation [5, 12]. 5. SUMMARY AND CONCLUSIONS The Galileo’s space-time-symmetry postulate is followed by some mutually matched corollaries: - the universal speed limit is equal for all material bodies in all inertial reference frames, - both momentum and mass of any particle are different in different reference frames, - any restless particle moves with the ultimate speed in any inertial reference frame, - the velocity of any unhurried particle is below the absolute speed limit in all reference frames, - elementary particles of the both kinds may be treated as wave packets. Thus, in agreement with [2, 3], the Galileo’s concept [6] may be regarded as an ancestor of the mod-ern relativistic and quantum theories. CONFLICTS OF INTEREST The authors declare no conflicts of interest regarding the publication of this paper. REFERENCES 1. Petelin, M. and Thumm, M. (2018) On the Evolution of Approaches to the Space-Time Symmetry. Natural Science, 10, 81-84. https://doi.org/10.4236/ns.2018.103008 2. Mermin, N.D. (1984) Relativity without Light. American Journal of Physics, 52, 119-124.  https://doi.org/10.1119/1.13917 3. Feigenbaum, M.J. (2008) The Theory of Relativity—Galileo’s Child. The Rockefeller University, New York. 4. Pauli, W. (1921) Theory of Relativity. Encyclopedia of Mathematical Sciences, B.G. Teubner, Leipzig. 5. Feynman, R.P., Leighton, R.B. and Sands, M. (2006) The Feynman Lectures on Physics. Addison-Wesley Long-man. 6. Drake, S. (1978) Galileo at Work—His Scientific Biography. University of Chicago Press, Chicago. 7. Petelin, M. (2015) The Universal Speed Limit as an Attribute of the Space-Time Symmetry. Journal of Basic and Applied Physics (JBAP), 4, 8-11. 8. Ritzfeld, A. (ed./Translation into German) (1912) Πρόκλου Διαδόχου Λυκίου, Στοιχείωσις Φυσική (Elements of Physics). Teubner, Leipzig.  9. Kimura, W.D. (2017) Electromagnetic Waves and Lasers. Morgan & Claypool Publishers, San Rafael.  https://doi.org/10.1088/978-1-6817-4613-5 10. Kar, D. (2019) Experimental Particle Physics. IOP Publishing Ltd., Bristol.  https://doi.org/10.1088/2053-2563/ab1be6 11. Einstein, A. (1905) Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 18, 639-643. https://doi.org/10.1002/andp.19053231314 12. Bohm, D. (1951) Quantum Theory. Prentice-Hall, Inc., New York. 

On the Elementary Mechanical Effects of the Space-Time-Symmetry Relativity

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On the Elementary Mechanical Effects of the Space-Time-Symmetry Relativity

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