What would happen to the signal of a theorically very efficient videocall (the signal travels at the light's speed), if this was between a spaceship, which travels at half the speed of light, and the earth, which for convenience we should consider steady? What would the man on the ship see? What would the man on the earth see? Ps. I'm pretty sure it has a known and quite simple explanation, I just can't get work it out.
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What would happen to the signal of a theorically very efficient videocall...
In the following, I assume the spacecraft is traveling away from the Earth. It's straightforward to change this to the traveling towards the Earth.
If the videocall signal is an analog transmission, the relativistic Doppler effect scales the frequency $f_c$ of the carrier and sidebands down by a factor of
$$f'_c = f_c\sqrt{\frac{1-\beta^2}{1+\beta^2}} = f_c\sqrt{\frac{1-0.5^2}{1+0.5^2}} \approx 0.775\cdot f_c$$
Thus, the spacecraft radio would need to tune to a lower frequency to receive the transmission than if it were at rest with respect to the transmitter.
Further, when the the signal is demodulated, the baseband signals are scaled by the same factor. For example, a middle A on a piano $(440\,\mathrm{Hz})$ would play back at about $341\,\mathrm{Hz}$ which is between an F and E below middle A.
So, to hear and see the baseband signal in its original form, the spacecraft receiver would need to buffer (record) the signal for some time and then play it back at a higher speed.
Similarly, the carrier for a digital transmission would be scaled down just as with the analog transmission so that the receiver tunes to a lower frequency. Further, the digital demodulater would need to be adjusted due to the change in sideband frequencies and bit rate.
As an aside, the receiver on Cassini for the Huygens probe was designed to follow the time varying Doppler shift of the carrier frequency as Cassini sped by Titan but, due to an engineering oversight, the digital decoder was not designed to track the change in bit rate and so, Cassini's path had to be altered to reduce the Doppler shift. For further reading see How Huygens avoided disaster
So, again, a buffer would be required to accumulate the digital data for some time so that it could be played back at the original bit rate.
Alfred Centauri
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