The observer starts moving towards the light source:
Two obvious facts in this particular scenario:
(A) The speed and the frequency of the light pulses vary proportionally for the observer.
(B) The wavelength (distance between light pulses) remains constant.
The fact (B) remains true in ANY scenario (moving emitter, moving observer, presence or absence of gravity) and will become the fundamental axiom of future, Einstein-free physics (if the death of physics is not irreversible):
The wavelength of light is constant (for a given emitter).
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 an absurdity.
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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