Types of Spacetimes  

General Types > s.a. causality conditions; causality-violating; Geodesically Complete; lorentzian geometry.
* Generic: Every timelike or null geodesic contains at least one point with nonzero Ka Kb K[c Rd]ab[e Kf ] (effective curvature).
* Strongly asymptotically predictable: Essentially, a spacetime with no naked singularities, except white holes; An asymptotically flat spacetime (M, g), with compactification (M', g'), such that in M there is a sufficiently large open region V (i.e., which contains the past of +, the closure of M' J(+) is in V), which makes (V', g') globally hyperbolic.
* Totally vicious: A spacetime (M, g) such that for all p in M, I(p) I +(p) = M, i.e., the Lorentzian distance function d(p, q) = ; Examples are a 2-torus with the usual metric, the Gödel spacetime, the Kerr-Newman solution with a2 + e2 > m2.
@ Generic: Beem & Harris GRG(93).
@ Totally vicious: Ikawa & Nakagawa JoG(88); Matori JMP(88); Kim & Kim JMP(93).
@ Related topics: Markowitz GRG(82) [conformally hyperbolic]; Hall CQG(96) [decompositions into constant-type regions]; Ramos et al JMP(03) [double warped, invariant characterization]; Pina & Tenenblat JGP(07) [conformally flat]; Senovilla CQG(08) [vanishing second covariant derivative of the Riemann tensor].
> In terms of curvature: see petrov types; riemann tensor [constant invariants]; Silent Universe; solutions; weyl tensor [purely magnetic].
> In terms of physical interpretation: see asymptotic flatness; black holes; cosmological models; Non-Imprisonment.
> Topologically non-trivial: see geons; wormholes.

Cylindrically Symmetric > s.a. axisymmetry; models in canonical general relativity; [cosmic strings].
* Idea: Axisymmetry with an additional z-invariance.
@ Levi-Civita: Rao JPA(71) [radiating]; Delice APPB(06)gq/04 [vacuum, non-static], gq/05-wd [radiating].
@ Other general relativity: Carot et al CQG(99)gq [def]; Arazi & Simeone GRG(00)gq; Barnes CQG(00)gq = gq/00-in; Senovilla & Vera CQG(00)gq [dust cosmology]; Klepac & Horsky CQG(00)gq [Einstein-Maxwell + fluid]; Qadir et al CQG(00)gq/07 [homotheties]; Fjällborg CQG(07) [static, Einstein-Vlasov]; Zofka & Bicak CQG(08)-a0712 [static, with cosmological constant].

Einstein Manifold / Metric / Space / Static Universe > s.a. embeddings; de sitter space.
$ Def: A manifold with Lorentzian metric such that Rab = c gab, with c a constant.
* Brinkman's theorem: Two Einstein spaces can be conformally mapped to each other only if both are Ricci-flat pp-waves, or both are conformally flat [@ Daftardar-Gejji GRG(98), with matter].
@ General references: Petrov 69; Besse 87; Gao JDG(90); Dancer & Strachan CQG(02)m.DG [on TSn+1]; Barrow et al CQG(03) [with pfluid, stability]; Gibbons et al CQG(04)ht [5D, on S3-bundle over S2]; Böhm JDG(04) [and simplicial complexes]; Boyer et al AM(05) [on spheres]; Mitra a0806/PRD; Kiosak & Matveev CRM(09)-a0905 [no conformal rescalings in complete case]; Carneiro & Tavakol PRD(09)-a0907 [stability in the presence of vacuum energy].
@ In other gravity theories: Seahra & Bohmer PRD(09)-a0901 [instability in f(R) theories]; Bohmer & Lobo PRD(09) [stability in modified Gauss-Bonnet gravity]; Miritzis PRD(09) [stability in fourth-order gravity]; Goheer et al CQG(09); > s.a. horava gravity.
@ Conformally Einstein: Gover m.DG/04-in [almost]; Gover & Nurowski JGP(06)m.DG/04 [obstructions, n-dimensional].

Stationary and Related Spacetimes > s.a. general relativity solutions with symmetries; Papapetrou Theorem.
* Ehlers group: A symmetry group of the vacuum Einstein equation for strictly stationary spacetimes.
* Pseudostationary: A spacetime with a one-parameter isometry group, such that its orbits are timelike curves at sufficiently large asymptotic distances (if they were timelike everywhere, the spacetime would be stationary).
@ Static: Bartnik & Tod CQG(06)gq/05 [and spatial 3-metrics]; Sánchez & Senovilla CQG(07) [global orthogonal decomposition]; Lafontaine JGP(09); > s.a. anti-de sitter spacetime; Ultrastatic Spacetimes.
@ Related topics: Mars CQG(01)gq [Ehlers group]; > s.a. spin coefficients.

Other Symmetries > s.a. general relativity solutions with symmetries; killing fields.
@ Flat: Barbot JGP(04)m.GT [globally hyperbolic]; Guediri DG&A(04) [compact]; Adler & Overduin GRG(05) [approximately flat]; Bonsante JDG(05) [with compact hyperbolic Cauchy surfaces]; > s.a. conservation laws; minkowski space.
@ 3D: Bona & Coll JMP(94) [with isometries]; Charette et al JGP(03) [with closed timelike curves].
@ Plane symmetric: Feroze et al JMP(01) [classification].
@ Homogeneous: Bueken & Vanhecke CQG(97) [curvature homogenous]; Valiente CQG(98)gq, CQG(99)gq/98, CQG(00) [polyhomogeneous]; Meessen LMP(06) [with canonically homogeneous null geodesics]; > s.a. bianchi models and FRW models.
@ Other spatial symmetries: Llosa & Carot a0907 [two Killing vector fields, flat deformations]; > s.a. Self-Similarity; spherical symmetry.

Other Specific Forms > s.a. coordinates [forms of line element]; Newman-Tamburino; W-Universe; Misner Metric; Misner Space.
* Generalized Lewis-Papapetrou: A metric of the form ds2 = –f (dt + i dxi)2 + f –1 gij dxi dx j ; > s.a. teleparallel gravity.
* Painlevé-Gullstrand: A metric written in the form ds2 = –[c2(r, t) – f 2(r, t)] dt2 – 2 f(r, t) dr dt + r2 d2.
@ Painlevé-Gullstrand: in Visser IJMPD(03)ht/01; Fischer & Visser AP(03)cm/02 [effective geometry]; Natário a0805 [for Kerr]; > s.a. schwarzschild.
@ Other types: Clarke & Joshi CQG(88) [reflecting]; > types of metrics [including singular, distributional].


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