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.

@ __References__: Senovilla GRG(98)-a1801;
Cotsakis LNP(02)gq [in cosmology];
Senovilla phy/06-conf;
Kulkarni a1911-BS [rev];
Senovilla REF-a2102.

**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, intros__: in Hawking & Ellis 73;
in Ryan & Shepley 75;
in Naber 88 [II];
Joshi 93;
Clarke 94 [III–IV];
in Witten a1905-ln.

@ __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];
Senovilla & Garfinkle CQG(15)-a1410 [rev, history];
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 the generalized second law];
Fewster & Galloway CQG(11)-a1012 [weakened energy conditions];
Bailleul JMP(11)-a1009 [probabilistic view];
Ishibashi & Maeda PRD(12)-a1208 [in asymptotically AdS spacetimes];
Graf et al CMP(18)-a1706 [for C\(^{1,1}\) Lorentzian metrics];
Lesourd CQG(18)-a1804 [with chronology violation in the interior];
Fewster & Kontou CQG(20)-a1907 [weakened energy hypotheses];
Lu et al JLMS-a1908 [weighted Lorentz-Finsler manifolds];
Minguzzi a1909
[compatible with quantum field theory and black hole evaporation];
Bittencourt et al EPJC(20)-a2003 [and disformal transformations].

@ __In other theories of gravity__: Low GRG(95);
Alani & Santillán JCAP(16)-a1602 [cosmological, for *f*(*R*) gravity];
Luz & Mena JMP(20)-a1909 [affine theories, with and without torsion];
> s.a. singularities.

@ __Generalized settings__: Aazami & Javaloyes CQG(16)-a1410 [in Finsler spacetime].

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