What Precisely Does "Preferred Foliation" Mean?

[inflector]

In another thread on dBB and nonlocality (http://www.physicsforums.com/showthread.php?p=2921342#post2919489), Demystifier wrote:
The only (currently known) way to avoid preferred foliation is the evolution with respect to a scalar parameter s.

What precisely does "preferred foliation" mean? I've seen the term used here in the QM section quite a bit but never explained. I tried googling it but didn't find any definitions just plenty of usage.

I have a very very rough idea what a foliation in mathematics means but don't understand the term "preferred foliation."

It seems to be something related to "preferred frame" and causal relationships over time. Any suggestions or pointers for further research would be greatly appreciated.

[Demystifier]

In a 3-dimensional world, foliation is a representation of the space as an infinite set of parallel (but possibly curved) 2-dimensional surfaces. Like an onion. In fact, any 3-dimensional object (like an apple) can be foliated in an infinite number of different ways. But for an onion, there is a preferred foliation - the foliation defined by the physical leafs themselves.

In the 4-dimensional world (associated with relativity) the idea is the same, but now the leafs are 3-dimensional hypersurfaces which can be thought of as spaces with constant time. The principle of relativity suggests that no foliation of 4-dimensional spacetime should be preferred. By contrast, the theories with a preferred foliation of spacetime suggest that that spacetime has a (hidden) onion structure that does not depend on the observer.

[inflector]

Thanks for the definition Demystifier.

In the 4-dimensional world (associated with relativity) the idea is the same, but now the leafs are 3-dimensional hypersurfaces which can be thought of as spaces with constant time.

Does this mean that time represents the transition between the leaves in the foliation represent the movement of time?

The principle of relativity suggests that no foliation of 4-dimensional spacetime should be preferred.

Does this mean that there would be different valid foliations existing at the same time and their validity would be equal under relativity in much the same way that there is no preferred frame in relativity?

By contrast, the theories with a preferred foliation of spacetime suggest that that spacetime has a (hidden) onion structure that does not depend on the observer.

What is it about the theories that causes them to require a preferred foliation? What are the implications of a theory having a preferred foliation?

Finally, is this related to the issues of quantum gravity since quantum mechanics seems to have a sense of fixed time while relativity does not?