The laws of physics are mostly time-symmetric - they work whether time runs backwards or forwards. However, we certainly perceive an "arrow of time" whenever we smash an egg or drop a wineglass - the egg and glass will not reassemble themselves. Linking this arrow of time defined by thermodynamics to the cosmological arrow of the expansion of the universe defines, for us, a forward direction.
Why does the arrow point that way? Could there be pockets in the universe where the arrow points in the opposite direction, where time seems, to us, to run backwards?
Lawrence Schulman of Clarkson University in New York has been tackling this question. He claims that it is possible to have opposite-running time regions and that some "small" degree of contact between them will not destroy the arrows of time. However, on the question of causality (whether someone in a time-forward region could warn someone in a time-backward region that an event is about to happen), Schulman admits that his calculations rely on some assumptions to avoid paradoxes.
How could regions of opposite-running time have formed? In the same way that electrons and positrons run in opposite time directions, Schulman suggests a cosmological picture with a timeline running between the Big Bang and the "Big Crunch" - the massive contraction marking the end of our universe. Then what would appear to us as time-reversed regions are just matter following the timeline in reverse.
Although current cosmological thinking disfavours the Big Crunch scenario, if it does happen, it is still certainly a very long time away. Therefore, if there are galaxies with backward-running time, following the timeline from Big Crunch to Big Bang, at this instant they are very old and hence not very luminous, although they are still exerting a gravitational pull - exactly the hallmark of dark matter. AIP