Both relativists and antirelativists believe that the motion of the light source does change the wavelength https://www.einstein-online.info/en/spotlight/doppler/
Stationary source: https://www.einstein-online.info/wp-content/uploads/SRT_Dopplereffekt_Pulse_%C2%A9_Daniela_Leitner_Markus_Poessel_Einstein-Online.gif
Moving source: https://www.einstein-online.info/wp-content/uploads/SRT_Dopplereffekt_Pulse_2_%C2%A9_Daniela_Leitner_Markus_Poessel_Einstein-Online-2.gif
It is easy to see that the wavelength varies with the speed of the source only if the speed of light is constant (independent of the speed of the source). However the speed of light is variable (c'=c+v), so the wavelength is constant in the moving-source case (I have tried to prove these claims elsewhere).
(frequency) = (speed of light)/(wavelength)
tolerates two axioms:
Axiom 1: The speed of light is constant.
Axiom 2: For a given emitter, the wavelength of light is constant.
Axiom 1 killed physics.
Axiom 2 will resurrect it (if it's not too late). Corollaries (established truths in future, Einstein-free physics):
Corollary 1: Any frequency shift entails (is caused by) a proportional speed-of-light shift.
Corollary 2: If the emitter and the observer (receiver) travel towards each other with relative speed v, the speed of light as measured by the observer is c' = c+v, as per Newton's theory.
Corollary 3: Spacetime and gravitational waves (ripples in spacetime) don't exist. LIGO's "discoveries" are fake.
Corollary 4: Light falls in a gravitational field with the same acceleration as ordinary falling bodies - near Earth's surface the accelerations of falling photons is g = 9.8 m/s^2. Accordingly, there is no gravitational time dilation: Einstein's general relativity is absurd.
Corollary 5: The Hubble redshift is due to light slowing down as it travels through vacuum. The universe is not expanding.
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