Spacetime Singularities  

In General
* Idea: A spacetime is said to be nonsingular if it is timelike and null geodesically complete, by analogy with Riemannian geometry, where geodesic completeness is equivalent to the usual metric completeness.
* Remark: When this condition is violated we may not have what we would like to call a singularity physically; And, when satisfied, we might have, e.g., timelike lines of finite acceleration which are incomplete.
* Consequences: Naked singularities would be a problem for predictive physics.

Trying to Avoid Singularities > s.a. quantum cosmology; quantum gravity, quantum gravity models and phenomenology.
* History: An attempt was made by the Soviets with the mixmaster universe; Other possibilities include cosmic censorship, violation of energy conditions, "gravastars", perhaps quantum gravity (non-commutative? > see Kasner Solutions).
@ General references: Einstein & Rosen BB(31), PR(35); Einstein AM(39), AM(45); Einstein & Straus AM(46); in Misner et al 73.
@ By violating energy conditions: Fulling & Parker PRD(73) [quantum]; Bekenstein PRD(75) [classical]; Fakir gq/98.
@ By going to a different metric: Quirós PRD(00)gq/99, et al PRD(00) [geometric duality in general relativity and Brans-Dicke]; Quirós gq/00, et al gq/00/PRD [conformal rescaling].
@ By extending the spacetime: Sniatycki in(91) [using Jacobi metric]; > s.a. schwarzschild
@ Related topics: Heller & Sasin IJTP(95), GRG(99)gq/98 [algebraically]; Raptis IJTP(06)gq/04 [Schwarzschild, finitary-algebraic]; Goswami & Joshi gq/05 [by not forming trapped surface]; Gershtein et al TMP(05)gq [in field theory of gravitation?]; Mac Conamhna a0708 [M-theory]; > s.a. modified em theory, non-commutative gravity.

Singularity Theorems
* Idea: They 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.
* 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 × R [@ Narita PRD(98)gq].
@ Generalizations: Borde PRD(94)gq, & Vilenkin IJMPD(96)gq [including inflation]; Raychaudhuri MPLA(00)gq; Senovilla gq/06-in, a0712-in [based on spatial averages].

Hawking & Penrose Theorem
* Idea: If a spacetime satisfies the chronology condition, the Einstein equation, the condition + 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].
@ 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).

Gannon's Theorem
* Idea: Any localized non-trivial topology in space will develop into a singularity.
@ References: Gannon JMP(75), GRG(76).

Other References > s.a. censorship; collapse [including Hoop conjecture]; cosmology and models; types; spacetime boundary.
@ Reviews: Canarutto RNC(88); Clarke in(88); Senovilla GRG(98); Cotsakis LNP(02)gq-in [in cosmology]; Rendall gq/05-in; Senovilla phy/06-in; Cotsakis gq/07-in.
@ Philosophical: Earman 95; Lam PhSc(07).
@ General: Geroch JMP(68), in(68); Hájícek GRG(70); Newman GRG(71); Penrose in(78); Barrow & Tipler PRP(79), PLA(81); Fuchs et al FdP(88); Natario m.DG/06 [introduction for mathematicians].
@ And initial surfaces: Wojtkiewicz PRD(90).
@ Data at the singularity: Eardley et al JMP(72); Tod CQG(90); & Goode & Wainwright.
@ Strength and physical properties: Kánnár & Rácz JMP(92); Kánnár GRG(95) [in Einstein-Cartan]; Kriele & Lim CQG(95); Ori PRD(00).
@ Role, uses of singularities: Earman FP(96); Lopez CQG(93); Horowitz & Myers GRG(95)gq.
@ Probing singularities: Horowitz & Marolf PRD(95)gq; Ishibashi & Hosoya PRD(99)gq; Piechocki PLB(02); Konkowski et al gq/04-in [quantum particles]; Blau et al JHEP(06)ht [with scalar fields].
@ In other theories: Eguchi MPLA(92) [topological field theories]; Novello et al CQG(00) [general relativity + non-linear electrodynamics]; > s.a. brans-dicke; 3D quantum gravity; quantum gravity phenomenology.

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