Causal Structure - Causal Structure

Causal Structure

For a point in the manifold we define

• The chronological future of, denoted, as the set of all points in such that chronologically precedes :

• The chronological past of, denoted, as the set of all points in such that chronologically precedes :

We similarly define

• The causal future (also called the absolute future) of, denoted, as the set of all points in such that causally precedes :

• The causal past (also called the absolute past) of, denoted, as the set of all points in such that causally precedes :

Points contained in, for example, can be reached from by a future-directed timelike curve. The point can be reached, for example, from points contained in by a future-directed non-spacelike curve.

As a simple example, in Minkowski spacetime the set is the interior of the future light cone at . The set is the full future light cone at, including the cone itself.

These sets defined for all in, are collectively called the causal structure of .

For a subset of we define

For two subsets of we define

• The chronological future of relative to , is the chronological future of considered as a submanifold of . Note that this is quite a different concept from which gives the set of points in which can be reached by future-directed timelike curves starting from . In the first case the curves must lie in in the second case they do not. See Hawking and Ellis.

• The causal future of relative to , is the causal future of considered as a submanifold of . Note that this is quite a different concept from which gives the set of points in which can be reached by future-directed causal curves starting from . In the first case the curves must lie in in the second case they do not. See Hawking and Ellis.

• A future set is a set closed under chronological future.
• A past set is a set closed under chronological past.

• An indecomposable past set is a past set which isn't the union of two different open past proper subsets.

• is a proper indecomposable past set (PIP).
• A terminal indecomposable past set (TIP) is an IP which isn't a PIP.

• The future Cauchy development of, is the set of all points for which every past directed inextendible causal curve through intersects at least once. Similarly for the past Cauchy development. The Cauchy development is the union of the future and past Cauchy developments. Cauchy developments are important for the study of determinism.

• A subset is achronal if there do not exist such that, or equivalently, if is disjoint from .

• A Cauchy surface is an closed achronal set whose Cauchy development is .

• A metric is globally hyperbolic if it can be foliated by Cauchy surfaces.

• The chronology violating set is the set of points through which closed timelike curves pass.

• The causality violating set is the set of points through which closed causal curves pass.

• For a causal curve, the causal diamond is (here we are using the looser definition of 'curve' whereon it is just a set of points). In words: the causal diamond of a particle's world-line is the set of all events that lie in both the past of some point in and the future of some point in .