Editing Penrose–Hawking singularity theorems (section)

=== Versions ===
There are many versions; below is the null version:

: Assume
# The [[null energy condition]] holds.
# We have a noncompact connected [[Cauchy surface]].
# We have a closed [[trapped null surface]] <math>\mathcal{T}</math>.
: Then, we either have null geodesic incompleteness, or [[closed timelike curve]]s.
:: ''Sketch of proof'': Proof by contradiction. The boundary of the future of <math>\mathcal{T}</math>, <math>\dot{J}(\mathcal{T})</math> is generated by null geodesic segments originating from <math>\mathcal{T}</math> with tangent vectors orthogonal to it. Being a trapped null surface, by the null [[Raychaudhuri equation]], both families of null rays emanating from <math>\mathcal{T}</math> will encounter caustics. (A caustic by itself is unproblematic. For instance, the boundary of the future of two spacelike separated points is the union of two future light cones with the interior parts of the intersection removed. Caustics occur where the light cones intersect, but no singularity lies there.) The null geodesics generating <math>\dot{J}(\mathcal{T})</math> have to terminate, however, i.e. reach their future endpoints at or before the caustics. Otherwise, we can take two null geodesic segments{{snd}}changing at the caustic{{snd}}and then deform them slightly to get a timelike curve connecting a point on the boundary to a point on <math>\mathcal{T}</math>, a contradiction. But as <math>\mathcal{T}</math> is compact, given a continuous affine parameterization of the geodesic generators, there exists a lower bound to the absolute value of the expansion parameter. So, we know caustics will develop for every generator before a uniform bound in the affine parameter has elapsed. As a result, <math>\dot{J}(\mathcal{T})</math> has to be compact. Either we have closed timelike curves, or we can construct a congruence by timelike curves, and every single one of them has to intersect the noncompact Cauchy surface exactly once. Consider all such timelike curves passing through <math>\dot{J}(\mathcal{T})</math> and look at their image on the Cauchy surface. Being a continuous map, the image also has to be compact. Being a [[timelike congruence]], the timelike curves can't intersect, and so, the map is [[injective]]. If the Cauchy surface were noncompact, then the image has a boundary. We're assuming spacetime comes in one connected piece. But <math>\dot{J}(\mathcal{T})</math> is compact and boundariless because the boundary of a boundary is empty. A continuous injective map can't create a boundary, giving us our contradiction.

:: ''Loopholes'': If closed timelike curves exist, then timelike curves don't have to intersect the ''partial'' Cauchy surface. If the Cauchy surface were compact, i.e. space is compact, the null geodesic generators of the boundary can intersect everywhere because they can intersect on the other side of space.

Other versions of the theorem involving the weak or strong energy condition also exist.