Singularity Theorems |

**In General**

* __Idea__: Theorems that establish
conditions under which a spacetime will develop a singularity.

* __History__: The existence
of singularities in general relativity has been known for a long time (e.g.,
Schwarzschild solution), but it was not until Penrose's work that it became
clear
that they are not an artifact of spherical symmetry in the collapse of
stars,
or of homogeneity and isotropy in the cosmological case, and are in
fact a
general phenomenon.

**Hawking & Penrose Theorem**

* __Idea__: If a spacetime
satisfies the chronology condition, the Einstein equation, the condition ρ +
3*p*_{i} > 0,
is sufficiently general, and admits a closed spacelike hypersurface, then it
cannot be geodesically complete along all timelike and null directions.

* __Remark__: If we want
to know where the singularity occurs, we have to add other, physically questionable,
conditions.

* __And space of metrics__:
The theorems imply that, if the set of Lorentz metrics on a manifold is given
a reasonable topology, those geodesically incomplete
are an open set.

@ __Texts__: in Hawking & Ellis 73; in Ryan & Shepley 75; in Naber
88 [II]; Joshi 93; Clarke 94 [III–IV].

@ __First hint__: Raychaudhuri PR(55).

@ __Theorem__: Hawking PRL(65);
Hawking & Ellis PL(65);
Penrose PRL(65);
Geroch PRL(66);
Hawking PRL(66),
PRS(66), PRS(66), PRS(67); Geroch AP(68),
in(70); Hawking & Penrose
PRS(70);
Khalatnikov & Lifshitz PRL(70);
Clarke CMP(76); Clarke & Schmidt GRG(77);
Tipler et al in(80); Clarke & Królak
JGP(85);
Borde CQG(87), JMP(87);
Szabados JMP(87);
Clarke pr(88); Ford IJTP(03)gq [rev,
quantum loopholes]; Kunzinger et al CQG(15)-a1411, CQG(15)-a1502 [for C^{1,1} metrics].

@ __And energy conditions__: Tipler PRD(78); Roman PRD(88); Parker & Wang
PRD(90).

@ __And causality violation__: Tipler PRL(76), AP(77);
Borde CQG(85);
Kriele
GRG(90),
PRS(90).

**Other Versions and Generalizations** > s.a. singularities [in other theories of gravity]; spacetime boundaries [abstract boundary].

* __Rigid singularity theorem__:
A globally hyperbolic spacetime satisfying the strong energy condition and
containing past trapped sets, either is timelike geodesically
incomplete
or splits isometrically as Σ × \(\mathbb R\) [@
Narita PRD(98)gq].

* __Gannon's theorem__:
Any localized non-trivial topology in space will develop into a singularity.

@ __Gannon's theorem__: Gannon JMP(75), GRG(76);
Costa e Silva CQG(10) [in higher dimensions].

@ __Versions and generalizations__: Borde PRD(94)gq,
& Vilenkin IJMPD(96)gq [including
inflation]; Raychaudhuri MPLA(00)gq;
Senovilla Pra(07)gq/06-in, EAS(08)-a0712 [based
on spatial averages]; Minguzzi CMP(09)
[either chronology violation or singularity]; Schleich & Witt a1006 [based
on the maximal Yamabe invariant, using topology and differentiable structure];
Galloway & Senovilla CQG(10) [based on trapped submanifolds of arbitrary codimension]; Wall CQG(13)-a1010 [from generalized second law]; Fewster & Galloway CQG(11)-a1012 [weakened energy conditions]; Bailleul JMP-a1009 [probabilistic view]; Ishibashi & Maeda PRD(12)-a1208 [in asymptotically AdS spacetimes]; Graf et al a1706 [for C\(^{1,1}\) Lorentzian metrics].

@ __In other theories of gravity__: Low GRG(95); Alani & Santillán JCAP-a1602 [cosmological, for *f*(*R*) gravity]; > s.a. singularities.

@ __Generalized settings__: Aazami & Javaloyes CQG-a1410 [in Finsler spacetime].

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