  But we know the answer to that question. A well-known theorem in general relativity says the following: any initial data set on a three dimensional manifold S with positive definite metric has a unique embedding into a maximal space-time in which S is a Cauchy surface (see e.g., Geroch and Horowitz 1979, p. 284 for more detail), i.e., there is a unique largest space-time which has S as a Cauchy surface and contains a consistent evolution of the initial value data on S. Now since S is a Cauchy surface this space-time does not have closed time like curves. But it may have extensions (in which S is not a Cauchy surface) which include closed timelike curves, indeed it may be that any maximal extension of it would include closed timelike curves. (This appears to be the case for extensions of states on certain surfaces of Taub-NUT space-times. See Earman, Smeenk, and Wüthrich 2003 in the Other Internet Resources). But these extensions, of course, will be consistent. So properly speaking, there are no constraints on states on space-like surfaces. Nonetheless the space-time in which these are embedded may or may not include closed time-like curves.