Past and future distinguishing

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:

Given are the definitions of these causality conditions for a Lorentzian manifold ${\displaystyle (M,g)}$. Where two or more are given they are equivalent.

Notation:

(See causal structure for definitions of ${\displaystyle \,I^{+}(x)}$, ${\displaystyle \,I^{-}(x)}$ and ${\displaystyle \,J^{+}(x)}$, ${\displaystyle \,J^{-}(x)}$.)

...
Wikipedia