Abstract
The first proof (Sections 2 - 4), applies Doppler shifts to the Fourier time sine series. It shows if K = f'/f (frequency ratio of the shift, f' the shifted frequency), then the value of the series at time = t occurs in the shifted series at time = t/K. That is because in each harmonic the sin(f't) and cos(f't) became sin(Kft) and cos(Kft). The original series could be the number of photons in area of a beam with encoded information. Therefore the number of observed photons and information has the same resultant Doppler shift as frequency. Resultant is total effect of axial, transverse and gravitational shifts. Mass and energy of light do not have the same Doppler shift which may indicate missing parts that Doppler shift and that mass traveling at the speed of light is different from other mass and energy. The second proof (Section 5) the vector equations of space time require observed time to have 3 dimensions if the speed of light is constant in all directions, but the Doppler shift in each direction is not the same. The blue shift (compression) of time has paradoxes. If time has many dimensions, that would solve the paradoxes, but break conservation laws. No solution of that is given here. It is not expected that radical a solution will have any followers.
Highlights
The analysis below applies Doppler shifts to the Fourier time trigonometric series
The first proof (Sections 2 - 4), applies Doppler shifts to the Fourier time sine series. It shows if K = f'/f, the value of the series at time = t occurs in the shifted series at time = t/K
The original series could be the number of photons in area of a beam with encoded information
Summary
The analysis below applies Doppler shifts (axial, transverse and gravitational) to the Fourier time trigonometric series. As far as the photon flux is concerned, they have a Fourier time trigonometric series of the function of F have the same value as the Doppler shifted (F') function at time = t/K as the original F(t) did at t. Note since the observed distance (in either time or space) between the photons changed not just the frequency of each because the observed axial time between changed This should be the same effects should be found in the energy transferred by a gravitational field as in the beam of photons above. Reference [4] appears to be slides for a course on accelerators for engineers and users posted online by the university, but the actual lectures may make mention of the Doppler effect of axial time between particles
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