Causality conditions

In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s. The weaker the causality condition on a spacetime, the more unphysical the spacetime is. Spacetimes with closed timelike curves, for example, present severe interpretational difficulties. See the grandfather paradox. It is reasonable to believe that any physical spacetime will satisfy the strongest causality condition: global hyperbolicity. For such spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface. There is a hierarchy of causality conditions, each one of which is strictly stronger than the previous. This is sometimes called the causal ladder. The conditions, from weakest to strongest, are: Non-totally vicious Chronological Causal Distinguishing Strongly causal Stably causal Causally continuous Causally simple Globally hyperbolic Given are the definitions of these causality conditions for a Lorentzian manifold . Where two or more are given they are equivalent. Notation: denotes the chronological relation. denotes the causal relation. (See causal structure for definitions of , and , .) For some points we have . There are no closed chronological (timelike) curves. The chronological relation is irreflexive: for all . There are no closed causal (non-spacelike) curves. If both and then Two points which share the same chronological past are the same point: For any neighborhood of there exists a neighborhood such that no past-directed non-spacelike curve from intersects more than once. Two points which share the same chronological future are the same point: For any neighborhood of there exists a neighborhood such that no future-directed non-spacelike curve from intersects more than once. For any neighborhood of there exists a neighborhood such that there exists no timelike curve that passes through more than once.