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I've seen foliation used in the context of "foliation of spacetime" here and elsewhere in papers and such. Generally defined in reference to a "sequence of spatial hypersurfaces." But I don't know what that means either.

Again, I can imagine what these terms mean because of the English language meaning of the words. But what do these mean specifically in reference to the physics of spacetime?

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If you want to solve equations of motion to describe the time evolution of a system, either classically or quantum mechanically, you need to impose initial condition at one point in time, and then under some conditions the entire evolution of the system (forward and backwards) is determined. This is the type of things physicists do all the time.

Now, general relativity is a theory of spacetime, so it is not clear that any spacetime manifold will have well-defined evolution of the sort I described, where the conditions at a spatial slice at one point in time (called Cauchy surface) determines the system everywhere. For that to be true there has to be a way to separate what is the time direction at every point in spacetime.

If this can be done you express the spacetime as a series of spatial slices which evolves in time (called foliation of spacetime), and you have now a problem which amounts to describing how those spatial slices evolve, which is a traditional initial value problem which physicists know and love. Manifolds for which this can be done are called globally hyperbolic, and those are the ones which are easier to discuss, though there are well-known examples of spacetimes which are not globally hyperbolic.

Once you find one way to do it, one "foliation" of spacetime, usually there are many other ways, but the difficulty is usually in finding one way that works everywhere (it is always possible to do that separation only in some region of spacetime, but that exercise is not useful since you want to predict what happens everywhere, at any point in time).

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But it should be stated that it can always be done on a finite subdomain of the spacetime, even if it cannot be done over the whole spacetime. –  Jerry Schirmer Jan 11 '11 at 17:18

The easiest answer is that it just means a surface of constant $t$.

Now, this isn't completely trivial like that, because, in general relativity, there are many possible choices of $t$ one can make. For instance, take the Minkowski 2-plane, with coordinates $(t,x)$.

Certainly, one foliation of this surface would be saying that you just pick every $t=$constant surface. But, you have an infinity of more exotic choices, such as $t=\sin (x)+$cosntant, where each value for the constant picks a different surface.

Why do we do this? The basic idea is to go and break up the notion of spacetime into a space with time evolution. This is especially relevant if we're doing something like setting up two black holes as an initial condition, and then seeing what happens when we let them interact with each other.

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