QUESTION 1: The frequency of light (as measured by the observer) varies with the speed of the observer. Does this mean that the speed of light (as measured by the observer) also varies with the speed of the observer, in violation of Einstein's special relativity?
"vO is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + vO. (...) The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."
"La variation de la fréquence observée lorsqu'il y a mouvement relatif entre la source et l'observateur est appelée effet Doppler. (...) 6. Source immobile - Observateur en mouvement: La distance entre les crêtes, la longueur d'onde lambda ne change pas. Mais la vitesse des crêtes par rapport à l'observateur change !"
Carl Mungan: "Consider the case where the observer moves toward the source. In this case, the observer is rushing head-long into the wavefronts... (...) In fact, the wave speed is simply increased by the observer speed, as we can see by jumping into the observer's frame of reference."
Roger Barlow, Professor of Particle Physics: "Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/(lambda) waves pass a fixed point. A moving point adds another vt/(lambda). So f'=(c+v)/(lambda)."
Tony Harker, University College London: "If the observer moves with a speed Vo away from the source (...), then in a time t the number of waves which reach the observer are those in a distance (c-Vo)t, so the number of waves observed is (c-Vo)t/lambda, giving an observed frequency f'=f((c-Vo)/c) when the observer is moving away from the source at a speed Vo."
Albert Einstein Institute: "As the receiver moves towards each pulse, the time until pulse and receiver meet up is shortened. In this particular animation, which has the receiver moving towards the source at one third the speed of the pulses themselves, four pulses are received in the time it takes the source to emit three pulses [that is, the speed of light as measured by the receiver is (4/3)c]."
QUESTION 2: The frequency of light falling in a gravitational field increases in accordance with the equation f'=f(1+gh/c^2), as predicted by Newton's emission theory of light. Does this mean that the speed of light also increases, in accordance with another prediction of the emission theory, c'=c(1+gh/c^2)?
University of Illinois at Urbana-Champaign: "Consider a falling object. ITS SPEED INCREASES AS IT IS FALLING. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, WE SHOULD OBSERVE THE SAME EFFECT FOR LIGHT. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction."
Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices."
The Gravitational Red-Shift, R.F.Evans and J.Dunning-Davies, Department of Physics, University of Hull: "Attention is drawn to the fact that the well-known expression for the red-shift of spectral lines due to a gravitational field may be derived with no recourse to the theory of general relativity. This raises grave doubts over the inclusion of the measurement of this gravitational red-shift in the list of crucial tests of the theory of general relativity. (...) In truth, it would seem that the result for the red-shift of spectral lines due to the action of a gravitational field has nothing specifically to do with the theory of general relativity. It is a result which draws on more modern results due to such as Planck and Poincaré, but, apart from those, is deduced from notions of Newtonian mechanics alone."
"Relativity 3 - gravity and light"
Harvey Reall, University of Cambridge: "...light falls in the gravitational field in exactly the same way as a massive test particle."
Dr. Cristian Bahrim: "If we accept the principle of equivalence, we must also accept that light falls in a gravitational field with the same acceleration as material bodies."
"Le principe d'équivalence, un des fondements de base de la relativité générale prédit que dans un champ gravitationnel, la lumière tombe comme tout corps matériel selon l'acceleration de la pesanteur."
Robert W. Brehme: "Light falls in a gravitational field just as do material objects."