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
Famous quotes containing the word discussion:
“Americans, unhappily, have the most remarkable ability to alchemize all bitter truths into an innocuous but piquant confection and to transform their moral contradictions, or public discussion of such contradictions, into a proud decoration, such as are given for heroism on the battle field.”
—James Baldwin (1924–1987)
“What chiefly distinguishes the daily press of the United States from the press of all other countries is not its lack of truthfulness or even its lack of dignity and honor, for these deficiencies are common to the newspapers everywhere, but its incurable fear of ideas, its constant effort to evade the discussion of fundamentals by translating all issues into a few elemental fears, its incessant reduction of all reflection to mere emotion. It is, in the true sense, never well-informed.”
—H.L. (Henry Lewis)
“We cannot set aside an hour for discussion with our children and hope that it will be a time of deep encounter. The special moments of intimacy are more likely to happen while baking a cake together, or playing hide and seek, or just sitting in the waiting room of the orthodontist.”
—Neil Kurshan (20th century)