Is Einstein's theory about the fabric of space directly related to the doppler effect?

Question

If you are traveling at the speed of light, would you "push" light (matter) to a higher freq enabling you to skim across the high points like skipping a rock across a pond?

Answer 1

Einstein's theory about the fabric of space is mostly associated with gravity and the effects a mass has on the space which surrounds it.
On the other hand, the Doppler effect is concerned with how frequency is changed depending on whether the source is moving toward or away from an observer.
Your ideas of "pushing" the frequency is probably feasible if we consider only classical mechanics - but that isn't the case here. Once you obtain speeds close to that of light, time and distance is shortened (dilated). Since you would obviously measure the frequency by wave length and time - it would not be "pushed" because the speed of light is constant to every observer regardless of their relative motion.
Not very intuitive because of our observations on earth with speeds that are a very small percentage of light, but true none the less at speeds that are a significant percentage of light.

Answer 2

I don't know about the implications of 'fabric of space' in that question, but...

Yes, the observation of this effect is what lead to the theory of the expansion of the universe. Galaxies moving quickly away from us emit lower energy (red tinted) light and those moving toward us emit high energy (blue tinted) light. The Doppler effect is a phenomenon which applies to all waves, including electromagnetic.

Andromeda appears slightly blue, we know its heading right for us, and thankfully none of us will be alive when we meet.