Editing Penrose–Hawking singularity theorems (section)

=== Tools employed ===
A key tool used in the formulation and proof of the singularity theorems is the [[Raychaudhuri equation]], which describes the divergence <math>\theta</math> of a [[congruence (general relativity)|congruence]] (family) of geodesics. The divergence of a congruence is defined as the derivative of the log of the determinant of the congruence volume. The Raychaudhuri
equation is

:<math>\dot{\theta} = - \sigma_{ab}\sigma^{ab} - \frac{1}{3}\theta^2 - {E[\vec{X}]^a}_a</math>

where <math>\sigma_{ab}</math> is the shear tensor of the congruence and <math>{E[\vec{X}]^a}_{a} = R_{mn} \, X^m \, X^n</math> is also known as the Raychaudhuri scalar (see the [[congruence (general relativity)|congruence]] page for details). The key point is that <math>{E[\vec{X}]^a}_a</math> will be non-negative provided that the [[Einstein field equations]] hold and<ref name="penrose_hawking"/>

* the [[Energy condition#Null energy condition|null energy condition]] holds and the geodesic congruence is null, or
* the [[Energy condition#Strong energy condition|strong energy condition]] holds and the geodesic congruence is timelike.

When these hold, the divergence becomes infinite at some finite value of the affine parameter. 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'''.

This is relevant for singularities thanks to the following argument:

# Suppose we have a spacetime that is [[globally hyperbolic]], and two points <math>p</math> and <math>q</math> that can be connected by a [[Timelike curve|timelike]] or [[null curve]]. Then there exists a geodesic of maximal length connecting <math>p</math> and <math>q</math>. Call this geodesic <math>\gamma</math>.
# The geodesic <math>\gamma</math> can be varied to a longer curve if another geodesic from <math>p</math> intersects <math>\gamma</math> at another point, called a [[conjugate point]].
# From the focusing theorem, we know that all geodesics from <math>p</math> have conjugate points at finite values of the affine parameter. In particular, this is true for the geodesic of maximal length. But this is a contradiction{{snd}}one can therefore conclude that the spacetime is geodesically incomplete.

In [[general relativity]], there are several versions of the '''Penrose–Hawking singularity theorem'''. Most versions state, roughly, that if there is a [[trapped null surface]] and the [[Stress–energy tensor|energy density]] is nonnegative, then there exist [[geodesic]]s of finite length that cannot be extended.<ref>{{cite web |url=http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&page=articlesu7.html |title=Gravitational Lensing from a Spacetime Perspective |archive-url=https://web.archive.org/web/20070301163933/http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&page=articlesu7.html |archive-date=2007-03-01}}</ref>

These theorems, strictly speaking, prove that there is at least one non-spacelike geodesic that is only finitely extendible into the past but there are cases in which the conditions of these theorems obtain in such a way that all past-directed spacetime paths terminate at a singularity.