Description of Lorentz Transformations, the Doppler Effect, Hubble's Law, and Related Phenomena in Curvilinear Coordinates by Generalized Biquaternions
Abstract
This paper presents the derivation of Lorentz transformations in curvilinear coordinates utilizing generalized biquaternions. Generalized biquaternions are rotations in curvilinear coordinates, including on the <i>tx, ty</i>, and <i>tz</i> planes. These space-time rotations are precisely the Lorentz transformations in curvilinear coordinates. The orbital rotation of the source and/or receiver, which mathematically represents the Lorentz transformation in spherical coordinates, is identified as the cause of the transverse Doppler effect. The change in wave frequency, specifically the "redshift," results in nonlinearities of Hubble's law manifesting as phenomena such as accelerated and anisotropic expansion of the universe, aberration, and wave polarization. Apparently, redshift exists even without radial expansion of the universe, i.e., without the "Big Bang". The reasons for the accelerated expansion of the universe, the anisotropic (angular) distribution of relic radiation, and the polarization of light from distant stars become clear in this approach. This greatly simplifies the mathematical description and understanding of the supposedly complex processes occurring in the universe.