Since time slows down and length contracts, when we travel almost at speed of light, if the speed of light (or EM waves) remains same and the wavelength of light remains same, do we measure the wavelength more than it actually is? Does it mean, we can "see" the ultraviolet light, if we can contract our length by an order of 100..?
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3 Answers
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Yes, though it isn't quite as you think.
You've probably heard of the Doppler effect. Even without the help of relativity we can see ultraviolet light if the source is moving away from us fast enough because the Doppler effect reduces the frequency of the light. We still get this effect in special relativity and it's called the relativistic Doppler effect.
If the source of the ultraviolet light is moving directly towards us then the frequency of the light is increased not decreased, so we can't see it. However if the source is moving towards us but not directly towards us then how the light is red/blue shifted depends on how fast the source is moving and how far we are from the source's line of travel. The article I linked above gives some details though I'm afraid calculating this is quite involved.
answered Apr 2, 2013 at 11:35
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I'd say that it depends on how much velocity you're traveling relative to an observer who is gonna measure the light emitted by you.
This has already been a done-deal, which is so called the Relativistic Doppler Shift after resolving through spacetime diagrams via SR's postulates, by which we can measure the apparent frequency higher (blue-shift) or lower (red-shift) as the source moves towards or away from us.
answered Apr 2, 2013 at 11:37
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John Rennie has given a nice answer explaining that what you were looking for was really the Doppler effect. This answer is intended to complement John's by explaining why your original attempt at analysis was giving an incorrect result (predicting a redshift regardless of the direction of motion). The basic problem is that you can't apply the concepts of length contraction and time dilation to a light wave.
Length contraction is an effect in which an object is measured to be longest in a frame in which the object is at rest and is contracted in other frames byt a factor of $\gamma$. There isn't any frame in which a light wave is at rest, so it doesn't make sense to talk about length contraction of its wavelength.
The same issue prevents us from talking about time dilation of a light wave's frequency.
In general, it's not possible to reduce all of relativity to an analysis of length contractions and time dilations. You really need the Lorentz transformation, which includes other effects such as phase shifts of moving clocks.