Actually, this isn't really about when time stops, more about
when a large fraction of how we perceive time disappears.
According to this article: "Cosmos to freeze-frame" from Nature
Science Update: http://www.nature.com/nsu/011220/011220-4.html
there is a not so insignificant problem to be caused in about
50 billion years due to the accelerating expansion of the Universe.
Essentially our perception of other galaxies will go into
a freeze frame mode and then gradually fade to black. After
100 billion years we will only be able to see ~1000 galaxies.
This is pretty significant IMO, because stars less than around
0.3 solar masses (~M4-5) have lifetimes that may be this long.
Even the sun might make it if we do some star lifting and
use our energy resources wisely.
So an interesting question will be whether it will be
feasible for advanced civilizations to navigate themselves
at sufficiently fast velocities to overcome the expansion
or whether, in the end, there will be only one...
Robert
Ref:
Loeb, A. The long-term future of extragalactic astronomy.
Physical Review D, (2001).
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