Causal hierarchy

The causal hierarchy orders restrictions on the causal structure of a spacetime, from exclusion of closed timelike curves through stable causality and global hyperbolicity.

The chronological future $I^{+}(S)$ of a subset $S$ of a spacetime is the set of points reachable from $S$ by future-directed timelike curves. The chronological future is an open set.

A subset $U$ is chronologically saturated in both directions when $I^{+}(U)=U=I^{-}(U)$. If the ambient spacetime is connected and $U$ is nonempty, then $U$ is the whole spacetime.

The chronological past $I^{-}(S)$ is the set of points from which a future-directed timelike curve reaches $S$. Equivalently, it is the time reverse of the chronological future.

A closed timelike curve is a closed curve whose tangent is everywhere timelike, so an observer following it returns to the same spacetime event.

A spacetime is stably causal when its light cones can be widened slightly without creating a closed timelike curve. Equivalently, it admits a continuous time function strictly increasing along every future-directed causal curve.

Strong causality means that every event has arbitrarily small neighborhoods which no causal curve can leave and then re-enter. Stable causality implies strong causality.

A spacetime is future-distinguishing when $I^{+}(p)=I^{+}(q)$ implies $p=q$. Every strongly causal spacetime is future-distinguishing.

A globally hyperbolic spacetime has no causal pathologies and admits a Cauchy hypersurface intersected exactly once by every inextendible causal curve.

An achronal set contains no pair of points connected by a timelike curve.

An achronal boundary is the boundary of a chronological future or past. Its null generators cease to belong to the boundary after developing a conjugate point or intersecting another generator in the relevant way.

A Cauchy hypersurface is an achronal hypersurface met exactly once by every inextendible timelike curve. Its topology records the global spatial topology of a globally hyperbolic spacetime.

The future or past Cauchy horizon of a partial Cauchy hypersurface is the boundary of the region whose physical evolution is determined by data on that hypersurface. Crossing it can destroy uniqueness of the maximal globally hyperbolic development.