The Question

Galaxies at the fartherst reaches of the universe are travelling at near the speed of light correct? Does this mean that time within those galaxies is at a slower rate?

The Answer

Yes, from our point of view. From their point of view, time is moving slower in our galaxy. The situation is completely symmetric. We can see this by imagining that there is a clock in each galaxy sending out regular tick signals. We specify that the clocks are constructed exactly the same. We can count the ticks, compare it with our clock, and see how time is changing in the other galaxy.

Well, there are such clocks: atoms of various elements that have definite frequencies at which they emit light. A frequency is just a number of ticks per second, and all atoms of a given kind, say sodium, have the same characteristic frequencies. Looking at a distant galaxy, we can see that their sodium atoms run slow, the light is redder than it should be. They see the same thing, our sodium atoms look redder to them than theirs.

In addition to the time dilation effect, distant galaxies show the Doppler effect of reddening because they are travelling away from us. The time dilation effect exists for any direction of relative motion.

The fact that each observer sees the other's clock running slow appears to be a paradox because of our limited experience with high speed. But that is just the way things are. For more information on the apparent paradox, see

http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/971109a.html

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