Canadian09 said:
If the supercluster is moving in one overall direction (X+) at a high velocity (V) and I'm traveling to a star that is technically behind our system (in terms of direction of velocity) at 10V, therefore moving X-, would time pass faster for me than in the two systems?
No. At least, not with the obvious interpretation of "time passing", which would be the time it takes you to complete the journey, according to your clock, compared to the time it takes according to clocks at rest in the supercluster--I'm assuming the stars you are traveling between are also at rest in the supercluster. But that's not the only possible interpretation; see below.
It's unfortunate that pop science presentations of relativity reduce all the complexities involved in these scenarios down to one rule of thumb: "moving faster slows time down". This rule happens to work in some very simple scenarios, but it stops working as soon as you go beyond those simple scenarios. The question in your OP assumes that you can apply this rule in all scenarios; but you can't.
The problem with the rule is illustrated by Orodruin's comment in post #10: there is no such thing as "moving" in any absolute sense. "Moving" is always relative. So in order to even apply the above rule of thumb, you have to define what things are "moving" relative to. And in the general case, there is no unique answer to this question; there is no unique choice of something that is "at rest" to which all "motion" can be referred.
It just so happens that in certain special cases, such as the Hafele-Keating experiment referred to in post #4, you can uniquely pick out something that can be considered "at rest" for purposes of analyzing just that special case. In the case of the H-K experiment, the thing that can be considered uniquely "at rest" is an idealized non-rotating Earth whose center of mass follows the same path through spacetime as the center of mass of the actual Earth. This defines an inertial frame to sufficient precision for analyzing the experiment; and we can use motion relative to this inertial frame as our definition of "motion" in that analysis. By that definition, the westbound clock moves slower than the clock on the (rotating) Earth's surface, and the eastbound clock moves faster; so the westbound clock will have more elapsed time than the clock on the (rotating) Earth's surface, and the eastbound clock will have less.
(Actually, in the real experiment, there is also gravitational time dilation due to altitude, as PeroK mentioned. I've left that out of the analysis I just gave because it happens not to change the relative ordering of the clock rates in this case, though of course it does affect the precise numbers.)
In the case of moving between stars in a supercluster, as I said above, there is an obvious interpretation of "moving" according to which the supercluster is at rest and you, in your ship, are "moving", so you have less elapsed time. But there are also other possible interpretations of what is "moving", and in this scenario, because you don't set out from and return to the same place (note that in the H-K experiment, all three clocks start out and end up co-located), there is no unique way to pick one interpretation of what is "moving" as the "right" one.
