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 + 3pi > 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 C1,1 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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