Editing Penrose–Hawking singularity theorems (section)

== Elements of the theorems ==
{{Unreferenced section|date=December 2008}}
In history, there is a deep connection between the curvature of a [[manifold]] and its [[topology]]. The [[Bonnet–Myers theorem]] states that a complete [[Riemannian manifold]] that has [[Ricci curvature]] everywhere greater than a certain positive constant must be [[Compact space|compact]]. The condition of positive Ricci curvature is most conveniently stated in the following way: for every geodesic there is a nearby initially parallel geodesic that will bend toward it when extended, and the two will intersect at some finite length.

When two nearby parallel [[geodesic]]s intersect (see [[conjugate point]]), the extension of either one is no longer the shortest path between the endpoints. The reason is that two parallel geodesic paths necessarily collide after an extension of equal length, and if one path is followed to the intersection then the other, you are connecting the endpoints by a non-geodesic path of equal length. This means that for a geodesic to be a shortest length path, it must never intersect neighboring parallel geodesics.

Starting with a small sphere and sending out parallel geodesics from the boundary, assuming that the manifold has a [[Ricci curvature]] bounded below by a positive constant, none of the geodesics are shortest paths after a while, since they all collide with a neighbor. This means that after a certain amount of extension, all potentially new points have been reached. If all points in a connected manifold are at a finite geodesic distance from a small sphere, the manifold must be compact.

Roger Penrose argued analogously in relativity. If [[null geodesic]]s, the paths of [[ray (optics)|light rays]], are followed into the future, points in the future of the region are generated. If a point is on the boundary of the future of the region, it can only be reached by going at the speed of light, no slower, so null geodesics include the entire boundary of the proper future of a region.{{Citation needed|date=December 2008}} When the null geodesics intersect, they are no longer on the boundary of the future, they are in the interior of the future. So, if all the null geodesics collide, there is no boundary to the future.

In relativity, the Ricci curvature, which determines the collision properties of geodesics, is determined by the [[energy–momentum tensor|energy tensor]], and its projection on light rays is equal to the null-projection of the energy–momentum tensor and is always non-negative. This implies that the volume of a [[congruence (general relativity)|congruence]] of parallel null geodesics once it starts decreasing, will reach zero in a finite time. Once the volume is zero, there is a collapse in some direction, so every geodesic intersects some neighbor.

Penrose concluded that whenever there is a sphere where all the outgoing (and ingoing) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge.<ref>{{Cite book |last1=Hawking |first1=S. W. |name-list-style=amp |last2=Ellis |first2=G. F. R. |date=1994 |title=The Large Scale Structure of Space Time |location=Cambridge |publisher=[[Cambridge University Press]] |isbn=0-521-09906-4 }}</ref> This is significant, because the outgoing light rays for any sphere inside the horizon of a [[black hole]] solution are all converging, so the boundary of the future of this region is either compact or comes from nowhere. The future of the interior either ends after a finite extension, or has a boundary that is eventually generated by new light rays that cannot be traced back to the original sphere.