There is an object in the universe for which the Sun appears to be more stationary as it travels through the universe than a star on some other system. There is probably another object for which both our sun and the 2nd star (combined) appear to be more stationary. Does there exist in an object in the universe for which all objects appear to be the most stationary (on average)? What about an object that makes all other objects appear to be the least stationary? Would identifying this object have any significance at all or is it just a random rock somewhere in space that has some coordinates that will fail to offer additional insight? Wouldn't this object experience the least amount of time dilation (i.e. everything else is in slow motion with respect to this object)?
2 Answers
To make sense of this question you need to decide what an "object" means (is a rock an object or a conglomeration of a vast number of much smaller objects?).
Once you've settled that, you need to decide whether you're averaging velocities or speeds. If there are three objects, and two of them are moving away from me at the same speed $v$ in opposite directions, are those two objects, on average, stationary with respect to me or are they, on average, moving away from me at speed $v$? Since you want to talk about things like "the smallest", you presumably mean speed (which is a number) rather than velocity (which is not).
In a non-flat spacetime, there's also the issue of how you define the velocity relative to you of a very distant object with which you do not share a coordinate patch.
And finally, because velocities are always changing, we need to do this for all objects at a fixed time, which is going to require something like a global time coordinate.
But once you've come up with (necessarily pretty arbitrary) answers to all these questions and issues, the answer to your question is: It depends.
If there are finitely many objects, you can certainly do your averaging to associate a number to each one, and then choose the smallest of those numbers.
If there are infinitely many objects, it's less clear how to do the averaging. You'll need some kind of measure on the space of objects to integrate against. Once you figure out how to do that, you'll have associated a number to each object, and those numbers are bounded below (by zero) so they have a greatest lower bound. There might or might not be some object that realizes that bound, but there are certainly objects that get arbitrarily close.
Why you would ever want to do this is, of course, a much harder question.
Edited to add: Re your question about time dilation --- time dilation does not scale linearly with speed, so the object with the smallest average speed will probably not be the object with the smallest average time dilation factor. You could of course, in principle, identify that object --- but the reasons for doing so seem even more obscure than for the object with the smallest average speed.
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No, there cannot be a most stationary object in the universe. One of the basic tenets of relativity is that every inertial reference frame is equivalent. This is what Einstein said on the matter,
If a co-ordinate system $K$ be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates $K'$ which is subjected to an uniform translational motion relative to $K$.
(source)
If we have two objects, say a star & a comet, to continue with your astronomical objects, then we can create one coordinate system for the sun (call it $K$) and one for the comet (call it $K'$). In each of these coordinate systems, the respective object, sun or comet, is stationary (so the sun is stationary in $K$, the comet in $K'$). If $K$ and $K'$ differ (one sees the other moving), then the sun believes it is stationary and sees the comet as moving while the converse is true for the comet. And they're both right.
On the flip side, in order for these two frames to agree that the sun is stationary is for them to both be the same frame: $K'=K$. This is what physicists call co-moving (proper) frames.
You want to extrapolate this to find $n$ frames such that they are all described by coordinate system $C$. Given the $\sim10^{11}$ stars in the galaxy and (approximately) equal number of galaxies, all of which are moving in a variety of ways, so you will not be able to find such a 'most stationary' frame, it just doesn't exist.
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