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.^{[1]}
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.
The hierarchy
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:

Nontotally 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 (M,g). Where two or more are given they are equivalent.
Notation:
(See causal structure for definitions of \,I^+(x), \,I^(x) and \,J^+(x), \,J^(x).)
Nontotally vicious

For some points p \in M we have p \not\ll p.
Chronological
Causal

There are no closed causal (nonspacelike) curves.

If both p \prec q and q \prec p then p = q
Distinguishing
Pastdistinguishing

Two points p, q \in M which share the same chronological past are the same point:


I^(p) = I^(q) \implies p = q

For any neighborhood U of p \in M there exists a neighborhood V \subset U, p \in V such that no pastdirected nonspacelike curve from p intersects V more than once.
Futuredistinguishing

Two points p, q \in M which share the same chronological future are the same point:
I^+(p) = I^+(q) \implies p = q

For any neighborhood U of p \in M there exists a neighborhood V \subset U, p \in V such that no futuredirected nonspacelike curve from p intersects V more than once.
Strongly causal

For any p \in M there exists a neighborhood U of p such that there exists no timelike curve that passes through U more than once.

For any neighborhood U of p \in M there exists a neighborhood V \subset U, p \in V such that V is causally convex in M (and thus in U).

The Alexandrov topology agrees with the manifold topology.
Stably causal
A manifold satisfying any of the weaker causality conditions defined above may fail to do so if the metric is given a small perturbation. A spacetime is stably causal if it cannot be made to contain closed causal curves by arbitrarily small perturbations of the metric. Stephen Hawking showed^{[2]} that this is equivalent to:

There exists a global time function on M. This is a scalar field t on M whose gradient \nabla^a t is everywhere timelike and futuredirected. This global time function gives us a stable way to distinguish between future and past for each point of the spacetime (and so we have no causal violations).
Globally hyperbolic
Robert Geroch showed^{[3]} that a spacetime is globally hyperbolic if and only if there exists a Cauchy surface for M. This means that:

M is topologically equivalent to \mathbb{R} \times\!\, S for some Cauchy surface S (Here \mathbb{R} denotes the real line).
See also
References

^ E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes in H. Baum and D. Alekseevsky (eds.), vol. Recent developments in pseudoRiemannian geometry, ESI Lect. Math. Phys., (Eur. Math. Soc. Publ. House, Zurich, 2008), pp. 299–358, ISBN=9783037190517, arXiv:grqc/0609119

^ S.W. Hawking, The existence of cosmic time functions Proc. R. Soc. Lond. (1969), A308, 433

^ R. Geroch, Domain of Dependence J. Math. Phys. (1970) 11, 437–449
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