Abstract
Einstein's velocity addition formula ofspecial relativity (SR) defines a transformationϕ v of the ballB c of radiusc inR 3, representing all possible velocities in an inertial systemK, onto identical ballB ′c , which represents the velocities in another systemK′, moving with velocity v relative toK. Sinceϕ v maps the zero velocity ofB c into arbitrary vector v ofB ′c ,B c is homogeneous under all possibleϕ v.
A similar homogeneity of the unit ballB inL(G, H) under a set of mapsϕ a, a ∈B, arises also in theLine Transmission Theory (TLT) for a lossless line. HereL(G, H) is the space of all linear operators between Hilbert spacesG,H, representing the signals on the line in the two directions. The explicit form ofϕ a is obtained naturally in TLT.
IfG andH are considered as time and space, respectively, anda as a linear transformationG →H (the velocity v), then the losslessness of the line is equivalent to the conservation of the time-space interval, and the TLT technique easily gives the following SR-related results:
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•Time and space contraction formulas and general transformation fromK toK′ in the natural operator form.
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•A description of the amount of the relativistic non-simultaneity due to distance by the operator adjoint toa.
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•The Einstein's velocity addition formula is obtained without use of the space-time transformationK →K′.
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•The group of all automorphisms ofB c, corresponding to the relativistic addition of velocities is shown to be similar to the group of all conformal automorphisms ofB c. This similarity is obtained by transforming the usual velocities intosymmetric velocities, introduced here.
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Friedman, Y., Naimark, A. The homogeneity of the ball inR 3 and special relativity. Found Phys Lett 5, 337–354 (1992). https://doi.org/10.1007/BF00690591
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DOI: https://doi.org/10.1007/BF00690591