The criterion for self-consistently translational motion of reference frames in universal kinematics

Ya. I. Grushka

Abstract


Universal kinematics as mathematical objects may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of the Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborhood of our Solar System. The present paper is devoted to investigation of self-consistently translational motion of reference frames in abstract universal kinematics. In the case of self-consistently translational motion we can give the clear and unambiguous definition of displacement as well average and instantaneous speed of the reference frame. Hence the uniform rectilinear motion is the particular case of self-consistently translational motion. So, the investigation of self-consistently translational motion is technically necessary for definition of classes of inertially-related reference frames (being in the state of uniform rectilinear mutual motion) in universal kinematics. In the paper we establish the necessary and sufficient condition on coordinate transformation between reference frames of universal kinematics assuring self-consistently translational motion of one reference frame relatively to another:

Theorem. Let $\mathcal{F}$ be vector universal kinematics.
The reference frame $\mathfrak{m}\in\mathcal{L}k\left(\mathcal{F}\right)$ is self-
consistently translational relatively the reference frame
$\mathfrak{l}\in\mathcal{L}k\left(\mathcal{F}\right)$ if and only if, there exist the functions:
$\Phi:\mathbb{M}k\left(\mathfrak{m}\right)\longmapsto\mathbf{Tm}\left(\mathfrak{l}\right)$,
$\mathbf{F}:\mathbf{Tm}\left(\mathfrak{l}\right)\longmapsto\mathbf{Zk}\left(\mathfrak{l}\right)$,
$\mathbf{g}:\mathbf{Zk}\left(\mathfrak{m}\right)\longmapsto\mathbf{Zk}\left(\mathfrak{l}\right)$,
satisfying the following conditions:
1. For each $\mathrm{x}\in\mathbf{Zk}\left(\mathfrak{m}\right)$ the function
$\Phi_{(\mathrm{x})}(t):=\Phi\left(t,\mathrm{x}\right)$,
$t\in\mathbf{Tm}\left(\mathfrak{m}\right)$ is a bijection between
$\mathbf{Tm}\left(\mathfrak{m}\right)$ and $\mathbf{Tm}\left(\mathfrak{l}\right)$.
2. The function $\mathbf{g}$ is a bijection between $\mathbf{Zk}\left(\mathfrak{m}\right)$
and $\mathbf{Zk}\left(\mathfrak{l}\right)$.
3. For every $\mathrm{w}\in\mathbb{M}k\left(\mathfrak{m}\right)$
the following equality is performed:
\[
\left[\mathfrak{l}{\leftarrow}\mathfrak{m}\right]\mathrm{w}=
\left(\Phi\left(t_{\mathrm{w}},\mathrm{x}_{\mathrm{w}}\right),
\mathbf{F}\left(\Phi\left(t_{\mathrm{w}},\mathrm{x}_{\mathrm{w}}\right)\right)
+\mathbf{g}\left(\mathrm{x}_{\mathrm{w}}\right)
\right),
\]
where $t_{\mathrm{w}}=\mathsf{tm}\left(\mathrm{w}\right)$,
$\mathrm{x}_{\mathrm{w}}=\mathsf{bs}\left(\mathrm{w}\right)$..


Keywords


universal kinematics; reference frames; self-consistently translational motion

References


Grushka Ya.I. (2014). Criterion of existence of universal coordinate transform in kinematic changeable sets. Bukovinian Mathematical Journal, 2(2-3), 59–71.

Grushka Ya.I. (2015). Kinematic changeable sets with given universal coordinate transforms. Proceedings of Institute of Mathematics NAS of Ukraine, 12(1), 74–118.

Grushka Ya.I. (2015). Evolutionary Extensions of Kinematic Sets and Universal Kinematics. Proceedings of Institute of Mathematics NAS of Ukraine, 12(2), 139–204.

Grushka Ya.I. (2015). Theorem on Evolutional Extension for Universal Kinematics. Bukovinian Mathematical Journal, 3(3-4), 67–77.

Grushka Ya.I. (2017). Draft introduction to abstract kinematics. (Version 2.0). Preprint: ResearchGate, 208 pages.

Baccetti Valentina, Tate Kyle, Visser Matt. (2012). Inertial frames without the relativity principle. J. High Energ. Phys., 2012(5), 43.

Baccetti Valentina, Tate Kyle, Visser Matt. (2012). Lorentz violating kinematics: Threshold theorems. J. High Energ. Phys., 2012(3), 28.

Baccetti Valentina, Tate Kyle, Visser Matt. (2013). Inertial frames without the relativity principle: breaking Lorentz symmetry. In Proceedings of the Thirteenth Marcel Grossmann Meeting on General Relativity, pages 1189–1191. World Scientific.

Mamone-Capria Marco. (2016). On the Fundamental Theorem of the Theory of Relativity. Foundations of Physics, 46(12), 1680–1712.

Berzi Vittorio, Gorini Vittorio. (1972). On space-time, reference frames and the structure of relativity groups. Ann. Inst. H. Poincaré Sect. A (N.S.), 16, 1–22.

Gorini Vittorio. (1971). Linear kinematical groups. Communications in Mathematical Physics, 21(2), 150–163.

Lugiato Luigi A., Gorini Vittorio. (1972). On the Structure of Relativity Groups. Journal of Mathematical Physics, 13(5), 665–671.

Recami E., Olkhovsky V.S. (1971). About Lorentz transformations and tachyons. Lettere al Nuovo Cimento, 1(4), 165–168.

Recami E., Mignani. R. (1972). More about Lorentz transformations and tachyons. Lettere al Nuovo Cimento, 4(4), 144–152.

Recami E. (1986). Classical Tachyons and Possible Applications. Riv. Nuovo Cim., 9(6), 1–178.

Goldoni R. (1972). Faster-than- light inertial frames, interacting tachyons and tadpoles. Lettere al Nuovo Cimento, 5(6), 495–502.

Vieira Ricardo S. (2012). An Introduction to the Theory of Tachyons. Revista Brasileira de Ensino de Fisica, 34(3), 1–15.

Tangherlini Frank Robert. (2009). The Velocity of Light in Uniformly Moving Frames. Abraham Zelmanov Journal, 2, 44–110.

Medvedev S.Yu., Galamba I.F. (2012). About some consequences from the Tangherlini transformations. Uzhhorod University Scientific Herald. Series Physics, (31), 174–184.

Medvedev Sergey Yu. (2014). Superluminal Velocities in the Synchronized Space-Time. Progress in Physics, 10(3), 151–156.

Hassani Mohamed. (2015). Foundations of Superluminal Relativistic Mechanics. Communications in Physics, 24(4), 313–332.

Grushka Ya.I. (2017). Unanimously-translational motion of reference frames in universal kinematics. Bukovinian Mathematical Journal, 5(3-4).

Grushka Ya.I. (2012). Changeable sets and their properties. Reports of the National Academy of Sciences of Ukraine, (5), 12–18.

Grushka Ya.I. (2012). Primitive changeable sets and their properties. Mathematical Bulletin of Taras Shevchenko Scientific Society, 9, 52–80.

Grushka Ya.I. (2012). Visibility in changeable sets. Proceedings of Institute of Mathematics NAS of Ukraine, 9(2), 122–145.

Grushka Ya.I. (2013). Base changeable sets and mathematical simulation of the evolution of systems. Ukrainian Math. J., 65(9), 1198–1218.

Grushka Ya.I. (2014). Changeable sets and their application for the construction of tachyon kinematics. Proceedings of Institute of Mathematics NAS of Ukraine, 11(1), 192–227.

Grushka Ya.I. (2015). Coordinate transforms in kinematic changeable sets. Reports of the National Academy of Sciences of Ukraine, (3), 24–31.

Grushka Ya.I. (2017). On Universal Coordinate Transform in Kinematic Changeable Sets. Methods Funct. Anal. Topology, 23(2), 133–154.

Møller C. (1957). The theory of relativity. International series of monographs on physics. Clarendon Press, Oxford.

Misner C.W., Thorne K.S., Wheeler J.A. (1973). Gravitation. Part 1. W. H. Freeman, 1973.


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