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.
Spacetimes with closed timelike curves, for example, present severe interpretational difficulties.
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.
For each of the weaker causality conditions defined above, there are some manifolds satisfying the condition which can be made to violate it by arbitrarily small perturbations of the metric.
Stephen Hawking showed[2] that this is equivalent to: Robert Geroch showed[3] that a spacetime is globally hyperbolic if and only if there exists a Cauchy surface for