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As I understand it, the twin paradox was integral to Einstein's conclusion that nothing can travel faster than the speed of light.

Why wasn't there a similar conclusion for a hypothetical scenario where 2 blind musicians play the trumpet whilst moving away from each other at a speed close to the speed of sound. They would obviously hear a slower/lower tone (doppler shift) but at no point did someone suggest that time was slowing down.

If a spacecraft left earth accelerating up to 80% of the speed of light and views earth through a telescope and notices both a red shift and an appearance of time slowing down (as you would expect), then on the return journey there is a blue shift and time appears to speed up (as you would expect) why does time actually HAVE to be tied to the blue/red shift effect? Why is it different than the hypothetical scenario with the sound waves? Furthermore the time's apparent slowing down on earth on the journey to the star is "made up" by the time speeding up on the return journey isn't it?

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Essentially, time has to be involved for light since the measured speed of the red or blue shifted light doesn't change. In the case of sound, if the trumpet player moved faster than sound relative to the medium, the sound from the other trumpet player would never 'catch up'. But, even when the twin is moving away from earth arbitrarily close to c, the light from Earth not only 'catches up', the twin measures it to have speed c. –  Alfred Centauri Feb 28 at 2:14
    
Re: the first sentence: "Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested." (Wikipedia) That is, special relativity implies the correct interpretation of the "paradox," not the other way around. –  Chris White Feb 28 at 3:15
    
@ChrisWhite Hmm so what was the element of the twin paradox that implied that light must be traveling at the same speed regardless of the dopler shift? –  Mike S May 1 at 7:00

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You may be misinterpreting what is meant by "slow down" in special relativity. It is a fundamentally different sort of slowing down compared to what happens during doppler shift.

It is true that the frequency of a periodic signal will tend to be lower as the source is moving away, and higher than would otherwise be expected as it approaches. This works with both sound and light. It seems you know this.

However, in special relativity, the thing that is producing the signal is actually producing them less frequently. Actually, it doesn't have to be a signal in the usual sense. In the twin paradox, the space-traveling twin's heart beats slower, both on the outgoing and incoming trips. Time does not speed back up on the return trip to make up for the loss.

To apply this same idea to light, if you want to determine what frequency of light you'd detect on (stationary) Earth from a fast-moving light source, you need to first take this time dilation, as it's called, into account. Only after doing this would you then consider how the doppler effect would affect your reception.

Actually, the light source could be moving perpendicularly to our line of sight. In that case, there would still be time dilation, but no doppler effect. That is, you'd still detect a lower frequency, but it wouldn't be due to the doppler effect.

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