Discussion
If is a space-like surface (i.e., a collection of points such that every pair is space-like separated), then is the future of, which is all the points which can be reached from while going forward in time on curves which are timelike or null. Similarly, the past of, is the same thing going back in time.
When there are no closed timelike curves, and are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of are the same and both include . The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve.
When there are closed timelike curves, or even when there are closed non-spacelike curves, a Cauchy surface still determines the future, but the future includes the surface itself. This means that the initial conditions obey a constraint, and the Cauchy surface is not of the same character as when the future and the past are disjoint.
If there are no closed timelike curves, then given a partial Cauchy surface and if, the entire manifold, then is a Cauchy surface. Any surface of constant in Minkowski space-time is a Cauchy surface.
Read more about this topic: Cauchy Surface
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