Penrose shared half of the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity".
The Penrose theorem guarantees that some sort of geodesic incompleteness occurs inside any black hole whenever matter satisfies reasonable energy conditions.
Hawking's singularity theorem is for the whole universe, and works backwards in time: it guarantees that the (classical) Big Bang has infinite density.
It is still an open question whether (classical) general relativity predicts spacelike singularities in the interior of realistic charged or rotating black holes, or whether these are artefacts of high-symmetry solutions and turn into null or timelike singularities when perturbations are added.
[citation needed] In general relativity, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or spacetime stops being a manifold.
In the collapsing star example, since all matter and energy is a source of gravitational attraction in general relativity, the additional angular momentum only pulls the star together more strongly as it contracts: the part outside the event horizon eventually settles down to a Kerr black hole (see No-hair theorem).
The proof is somewhat constructive – it shows that the singularity can be found by following light-rays from a surface just inside the horizon.
But the proof does not say what type of singularity occurs, spacelike, timelike, null, orbifold, jump discontinuity in the metric.
One can extend general relativity to a unified field theory, such as the Einstein–Maxwell–Dirac system, where no such singularities occur.
The Bonnet–Myers theorem states that a complete Riemannian manifold that has Ricci curvature everywhere greater than a certain positive constant must be compact.
When two nearby parallel geodesics intersect (see conjugate point), the extension of either one is no longer the shortest path between the endpoints.
This implies that the volume of a congruence of parallel null geodesics once it starts decreasing, will reach zero in a finite time.
[5] 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.
The singularity theorems use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvatures.
Presumably, at the end of the geodesic the observer has fallen into a singularity or encountered some other pathology at which the laws of general relativity break down.
A key tool used in the formulation and proof of the singularity theorems is the Raychaudhuri equation, which describes the divergence
Thus all geodesics leaving a point will eventually reconverge after a finite time, provided the appropriate energy condition holds, a result also known as the focusing theorem.
Most versions state, roughly, that if there is a trapped null surface and the energy density is nonnegative, then there exist geodesics of finite length that cannot be extended.