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

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1. GALILEO’S REFERENCE FRAMES

Our methodical paper [1] was stimulated by the assumption of Refs. [2,3] that the number of independent 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 $\xi ,\tau $ or ${\xi}^{\prime},{\tau}^{\prime}$ . The measurement units being taken identical, the symmetry-reciprocity of the reference frames is met with the following equations [1]

$\begin{array}{l}\xi =\gamma {\xi}^{\prime}+\kappa {\tau}^{\prime}\hfill \\ \tau =\kappa {\xi}^{\prime}+\gamma {\tau}^{\prime}\hfill \end{array}\iff \begin{array}{l}{\xi}^{\prime}=\gamma \xi -\kappa \tau \hfill \\ {\tau}^{\prime}=-\kappa \xi +\gamma \tau \hfill \end{array}$ (1)

being left-right cyclic recurrent due to the interrelation between the constant coefficients $\gamma $ and $\kappa $

${\gamma}^{2}-{\kappa}^{2}=1$ . (2)

Under the following Minkowski change of variables [4,5]

$\stackrel{^}{\tau}=\text{i}\tau ,\text{\hspace{1em}}{\stackrel{^}{\tau}}^{\prime}=\text{i}{\tau}^{\prime}$ (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

${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$ . (2a)

According to (1), the relative velocity of the reference frames is

$\beta =\kappa /\gamma $ . (4)

Equation (4) and the Pythagoras condition (2), (2a) give

$\gamma =\frac{1}{\sqrt{1-{\beta}^{2}}}$ . (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 ${{\beta}^{\prime}}_{b}$ relative to the ${\xi}^{\prime},{\tau}^{\prime}$ reference frame, the body velocity relative to the $\xi ,\tau $ frame is

${\beta}_{b}=\frac{{{\beta}^{\prime}}_{b}+\beta}{1+\beta {{\beta}^{\prime}}_{b}}$ . (6)

Figure 1. Galileo’s inertial reference frames [1].

Figure 2. Mutual rotation of Cartesian coordinate systems [7] $\xi ,\stackrel{^}{\tau}$ and ${\xi}^{\prime},{\stackrel{^}{\tau}}^{\prime}$ .

It is natural to postulate that the body mass m and the body momentum (quantity of body motion) p compose a vector $\left[\stackrel{^}{m},p\right]$ similar to the time-coordinate vector $\left[\stackrel{^}{\tau},\xi \right]$ 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

$\begin{array}{l}p=\gamma {p}^{\prime}+\kappa {m}^{\prime}\hfill \\ m=\kappa {p}^{\prime}+\gamma {m}^{\prime}\hfill \end{array}\iff \begin{array}{l}{p}^{\prime}=\gamma p-\kappa m\hfill \\ {m}^{\prime}=-\kappa p+\gamma m\hfill \end{array}$ (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:

${\beta}_{b}={{\beta}^{\prime}}_{b}=1$ (8)

the particle mass and the particle momentum being interrelated with the formulas

$m=\left(\gamma +\kappa \right){m}^{\prime}=\left(\gamma +\kappa \right){p}^{\prime}=p\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\iff \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}^{\prime}=\left(\gamma -\kappa \right)m=\left(\gamma -\kappa \right)p={p}^{\prime}$ (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 ${\xi}^{\prime},{\tau}^{\prime}$ :

${{\beta}^{\prime}}_{b}=0,\text{\hspace{1em}}\text{\hspace{0.17em}}{{p}^{\prime}}_{0}=0,\text{\hspace{1em}}\text{\hspace{0.17em}}{m}^{\prime}={m}_{0}$,(10)

then, according to (7), the mass and the momentum of the body in another reference frame $\xi ,\tau $ are

$m=\gamma {m}_{0}$ (11)

$p=\beta m$ . (12)

Therefore, as it follows from (11) and (5), if the body velocity ${\beta}_{b}$ 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

${\beta}_{b},{{\beta}^{\prime}}_{b}\le 1$,(13)

corresponding to the 12^{th} 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 scalar product of the vectors $\left[\stackrel{^}{m},p\right]$ and $\left[\stackrel{^}{\tau},\xi \right]$ remains invariant:

$m\tau -p\xi ={m}^{\prime}{\tau}^{\prime}-{p}^{\prime}{\xi}^{\prime}$,(14)

which may be attributed to a phase of a relevant wave

${\text{e}}^{-i\left(m\tau -p\xi \right)}$ . (15)

3.1. Restless Particles

If a particle is restless, according to (8) and (9), the wave (15) takes the form

${\text{e}}^{-im\left(\tau -\xi \right)}={\text{e}}^{-i{p}^{\prime}\left({\tau}^{\prime}-{\xi}^{\prime}\right)}$,(16)

where the frequencies-propagators $m=p$, ${m}^{\prime}={p}^{\prime}$ are interrelated with the formulas (9) corresponding to the Doppler effect [9]. The waves (16) may be composed in a packet

$\Phi \left(\tau -\xi \right)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}{g}_{m}{\text{e}}^{-im\left(\tau -\xi \right)}\text{d}m}$ (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)

$\kappa \ll 1$,(18)

the invariant (14) reduces to

$m\tau -p\xi ={m}_{0}\left(\gamma \tau -\kappa \xi \right)\approx {m}_{0}\tau +{\theta}_{\kappa}$,(19)

where

${\theta}_{\kappa}={m}_{0}\left(\frac{{\kappa}^{2}}{2}\tau -\kappa \xi \right)$ . (20)

The corresponding waves (15) of a common resting mass ${m}_{0}$ may be composed in a packet

$\Psi \left(\xi ,\tau \right)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}{g}_{\kappa}{\text{e}}^{-i\left(m\tau -p\xi \right)}\text{d}\kappa}$,(21)

which, with account of (19), may be written in the form

$\Psi \approx \psi {\text{e}}^{-i{m}_{0}\tau}$,(22)

where the envelope

$\psi \left(\xi ,\tau \right)={\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}{g}_{\kappa}{\text{e}}^{-i{\theta}_{k}}\text{d}\kappa}$ (23)

slowly depends on time and coordinate. By using (20), one finds that $\psi \left(\xi ,\tau \right)$ satisfies the following parabolic equation

$\frac{\partial \psi}{\partial \tau}=-\frac{i}{2{m}_{0}}\frac{{\partial}^{2}\psi}{\partial {\xi}^{2}}$ . (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]

$\begin{array}{l}\xi =x\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{\xi}^{\prime}={x}^{\prime}\\ \tau =ct\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}{\tau}^{\prime}=ct\end{array}$ (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 $c=3\times {10}^{8}\text{m}/\text{s}$ [9]. The wave (16) may be written in the form ${\text{e}}^{-i\left(\omega t-kx\right)}$,where $\omega $ is the angular frequency and $k=\omega /c$ is the wave vector. In accordance to [5,9], the “usual” photon energy is $\hslash \omega $ and the “usual” photon momentum is $\hslash k$,where $\hslash $ is the Planck constant divided by 2π.

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) correspond with the famous Einstein’s formula
$E=M{c}^{2}$ [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 “symmetric” resting mass
${m}_{0}$ and the “practical” mass
${m}_{e}$ of the electron are linearly interrelated
${m}_{0}\hslash ={m}_{e}c$,the electron wave-packet equation (24) may be converted to a form corresponding to the Schrödinger equation [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 modern relativistic and quantum theories.

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 Longman.

[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) Πρóκλου Διαδóχου Λυκíου, Στοιχεíωσιç Φυσικη (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.