On some properties of Hassani transforms

Ya. I. Grushka

doi:10.30970/ms.57.1.79-91

Ya. I. Grushka Institute of Mathematics NAS of Ukraine Kyiv, Ukraine

DOI: https://doi.org/10.30970/ms.57.1.79-91

Keywords: Hilbert space;, universal kinematic;, relativity principle

Abstract

In the present paper, based on the ideas of Algerian physicist M.E. Hassani, the generalized
Hassani spatial-temporal transformations in real Hilbert space are introduced. The original
transformations, introduced by M.E. Hassani, are the particular cases of the transformations,
introduced in this paper. It is proven that the classes of generalized Hassani transforms do
not form a group of operators in the general case. Further, using these generalized Hassani
transformations as well as the theory of changeable sets and universal kinematics, the mathematically
strict models of Hassani kinematics are constructed and the performance of the relativity
principle in these models is discussed.

References

O.-M.P. Bilaniuk, V.K. Deshpande, E.C.G. Sudarshan, “Meta” Relativity, American Journal of Physics, 30 (1962), №10, 718–723. doi: 10.1119/1.1941773.

O.-M.P. Bilaniuk, E.C.G. Sudarshan, Particles beyond the light barrier, Physics Today, 22 (1969), №5, 43–51. doi: 10.1063/1.3035574.

E. Recami, V.S. Olkhovsky, About Lorentz transformations and tachyons, Lettere al Nuovo Cimento, 1 (1971), №4, 165–168. doi: 10.1007/BF02799345.

R. Goldoni, Faster-than-light inertial frames, interacting tachyons and tadpoles, Lettere al Nuovo Cimento, 5 (1972), №6, 495–502. doi: 10.1007/BF02785903.

E. Recami, Classical Tachyons and Possible Applications, Riv. Nuovo Cim., 9 (1986), №6, 1–178. doi: 10.1007/BF02724327.

S.Yu. Medvedev, On the possibility of broadening special relativity theory beyond light barrier, Uzhhorod University Scientific Herald. Ser. Phys., 18 (2005), 7–15.

J.M. Hill, J. Barry Cox, Einstein’s special relativity beyond the speed of light, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 468 (2012), 4174–4192. doi:10.1098/rspa.2012.0340.

Ya.I. Grushka, Tachyon generalization for Lorentz transforms, Methods Funct. Anal. Topology, 19 (2013), №2, 127–145.

M.E. Hassani, Foundations of superluminal relativistic mechanics, Communications in Physics, 24 (2014), №4, 313–332.

Ya.I. Grushka, Draft introduction to abstract kinematics. (Version 2.0). Preprint: ResearchGate, 2017. URL: https://doi.org/10.13140/RG.2.2.28964.27521.

Ya.I. Grushka, Kinematic changeable sets with given universal coordinate transforms, Zb. Pr. Inst. Mat. NAN Ukr., 12 (2015), №1, 74–118.

C. Møller, The theory of relativity. Clarendon Press, Oxford, 1957.

Ya.I. Grushka, Changeable sets and their application for the construction of tachyon kinematics, Zb. Pr. Inst. Mat. NAN Ukr., 11 (2014), №1, 192–227.

M.A. Naimark, Linear Representations of the Lorentz group, Oxford, Pergamon Press, 1964.

Ya.I. Grushka, Theorem of non-returning and time irreversibility of Tachyon kinematics, Progress in Physics, 13 (2017), №4, 218–228.

Ya.I. Grushka, Base changeable sets and mathematical simulation of the evolution of systems, Ukrainian Math. J., 65 (2014), №9, 1332–1353. doi: 10.1007/s11253-014-0862-6.

Ya.I. Grushka, Criterion of existence of universal coordinate transform in kinematic changeable sets, Bukovyn. Mat. Zh., 2 (2014), №2-3, 59–71.

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

V. Baccetti, K. Tate, M. Visser, Inertial frames without the relativity principle, J. High Energ. Phys., 2012 (2012), №5, 43. doi: 10.1007/JHEP05(2012)119.

V. Baccetti, K. Tate, M. Visser, Lorentz violating kinematics: Threshold theorems, J. High Energ. Phys., 2012 (2012), №3, 28. doi: 10.1007/JHEP03(2012)087.

V. Baccetti, K. Tate, M. Visser, Inertial frames without the relativity principle: breaking Lorentz symmetry, Proceedings of the Thirteenth Marcel Grossmann Meeting on General Relativity, World Scientific, 2013, 1189–1191.

E.Di Casola, Sieving the landscape of gravity theories, SISSA, 2014.

S. Liberati, Tests of Lorentz invariance: a 2013 update, Classical Quantum Gravity, 30 (2013), №13, 133001, 50. doi: 10.1088/0264-9381/30/13/133001.

G. Shan, How to realize quantum superluminal communication. Preprint: arXiv, 1999. URL: https: //arxiv.org /abs/quant-ph/9906116.

A.L. Kholmetskii, O.V. Missevitch et al., The special relativity principle and superluminal velocities, Physics essays, 25 (2012), №4, 621–626. doi: 10.4006/0836-1398-25.4.621.

K.A. Peacock, Would superluminal influences violate the principle of relativity? Lato Sensu, revue de la Soci´et´e de philosophie des sciences, 1 (2014), №1, 49–62.

G.I. Burde, Cosmological models based on relativity with a privileged frame, International Journal of Modern Physics D, 29 (2020), №6. doi: 10.1142/S0218271820500388.