Final answer:
Passengers on board a spaceship moving at 0.90c away from Earth still see a green lamp due to being in the same reference frame as the light source. However, an observer on Earth will see the light as red due to the relativistic Doppler redshift, making the lamp appear red.
Step-by-step explanation:
The question pertains to the relativistic effects on light observed in different reference frames, specifically the Doppler shift in frequency due to motion at relativistic speeds. According to the theory of special relativity, when a spaceship is moving at a high speed relative to an observer, the frequency of light from a lamp on the spaceship will appear different to observers who are not moving with the spaceship.
In this scenario, the spaceship is moving away from Earth at 0.90 times the speed of light (0.90c). To passengers on board who are in the same inertial frame as the spaceship, they will observe no change in the frequency of light, meaning they will still see the lamp as emitting green light. This is because within the ship, the laws of physics are as if the ship is at rest.
However, to an observer on Earth, the light would be redshifted due to the relativistic Doppler effect because the spaceship is moving away from the observer. This stretching of the wavelength makes the light appear to shift towards the red end of the spectrum. Therefore, the answer is option 4, a red lamp.