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This article is cited in 2 scientific papers (total in 3 papers)
METHODOLOGICAL NOTES
Permutation asymmetry of the relativistic velocity addition law and non-Euclidean geometry
V. I. Ritus P. N. Lebedev Physical Institute, Russian Academy of Sciences
Abstract:
The asymmetry of the relativistic addition law for noncollinear velocities under the velocity permutation leads to two modified triangles on a Euclidean plane depicting the addition of unpermuted and permuted velocities and the appearance of a nonzero angle $\omega$ between two resulting velocities. A particle spin rotates through the same angle $\omega$ under a Lorentz boost with a velocity noncollinear to the particle velocity. Three mutually connected three-parameter representations of the angle $\omega$, obtained by the author earlier, express the three-parameter symmetry of the sides and angles of two Euclidean triangles identical to the sine and cosine theorems for the sides and angles of a single geodesic triangle on the surface of a pseudosphere. Namely, all three representations of the angle $\omega$, after a transformation of one of them, coincide with the representations of the area of a pseudospherical triangle expressed in terms of any two of its sides and the angle between them. The angle $\omega$ is also symmetrically expressed in terms of three angles or three sides of a geodesic triangle, and therefore it is an invariant of the group of triangle motions over the pseudo-sphere surface, the group that includes the Lorentz group. Although the pseudospheres in Euclidean and pseudo-Euclidean spaces are locally isometric, only the latter is isometric to the entire Lobachevsky plane and forms a homogeneous isotropic curved 4-velocity space in the flat Minkowski space. In this connection, relativistic physical processes that may be related to the pseudosphere in Euclidean space are especially interesting.
Received: April 14, 2008
Citation:
V. I. Ritus, “Permutation asymmetry of the relativistic velocity addition law and non-Euclidean geometry”, UFN, 178:7 (2008), 739–752; Phys. Usp., 51:7 (2008), 709–721
Linking options:
https://www.mathnet.ru/eng/ufn620 https://www.mathnet.ru/eng/ufn/v178/i7/p739
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Abstract page: | 513 | Full-text PDF : | 143 | References: | 74 | First page: | 1 |
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