A Fourier Series Proof That the Axial like the Transverse Optical Doppler Shift Impacts Time and Information Rates

Abstract

In writings on relativity time, the various relations are only changed by the transverse shift. This paper proves that the axial Doppler shift does that as well and gives some impacts of that on common differential relations in physics. When a modulated signal lasting a time = T is subjected to an optical Doppler shift K (either axial or transverse or both), where K is shifted frequency/original frequency, the Doppler shifted signal will last T/K. This because all shifted harmonics of its Fourier series (with a fundamental period of T) will last 1/K times the period of the original harmonic. The reader can graph any Fourier series and then graph its shifted series. The reader will see the shifted period is T/K. The Fourier series of the original repeats when time is greater than T and the shifted one when time is greater than T/K, which means the original series only represents the signal from time = 0 to T and the shifted series represents the shifted signal from time = 0 to T/K. Hence, the shifted one has all of the information in T/K as the original has in T. Therefore everything in the series including information is T/K long in the shifted series. Therefore, both the axial and the transverse Doppler shift change time periods in a vacuum, independent of material involved. That has not been obvious for over 100 years the axial shift changes time from the definition of frequency = 1/time.