Editing Penrose–Hawking singularity theorems (section)

== Nature of a singularity ==
The singularity theorems use the notion of geodesic incompleteness as a stand-in for the presence of infinite curvatures. Geodesic incompleteness is the notion that there are [[geodesic]]s, paths of observers through spacetime, that can only be extended for a finite time as measured by an observer traveling along one. 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.

=== Assumptions of the theorems ===
Typically a singularity theorem has three ingredients:<ref name="penrose_hawking">{{Cite book |first1=Stephen |last1=Hawking |name-list-style=amp |first2=Roger |last2=Penrose |title=The Nature of Space and Time |location=Princeton |publisher=[[Princeton University Press]] |date=1996 |isbn=0-691-03791-4 }}</ref>

# An [[energy conditions|energy condition]] on the matter,
# A condition on the [[global spacetime structure|global structure of spacetime]],
# Gravity is strong enough (somewhere) to trap a region.

There are various possibilities for each ingredient, and each leads to different singularity theorems.

=== 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.

=== 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.

=== Modified gravity ===
In modified gravity, the Einstein field equations do not hold and so these singularities do not necessarily arise. For example, in [[Infinite Derivative Gravity]], it is possible for <math>{E[\vec{X}]^a}_a</math> to be negative even if the Null Energy Condition holds.<ref>{{Cite journal|arxiv=1605.02080 |title=Defocusing of Null Rays in Infinite Derivative Gravity|journal=Journal of Cosmology and Astroparticle Physics|volume=2017|pages=017|last1=Conroy|first1=Aindriú|last2=Koshelev|first2=Alexey S|last3=Mazumdar|first3=Anupam|year=2016|issue=1|doi=10.1088/1475-7516/2017/01/017|bibcode=2017JCAP...01..017C|s2cid=115136697}}</ref><ref>{{Cite journal|arxiv=1705.02382 |title=Newtonian Potential and Geodesic Completeness in Infinite Derivative Gravity|journal=Physical Review D|volume=96|issue=4|last1=Conroy|first1=Aindriú|last2=Edholm|first2=James|year=2017|page=044012|doi=10.1103/PhysRevD.96.044012|bibcode=2017PhRvD..96d4012E|s2cid=45816145}}</ref>