Types of Spacetimes  

General Types > s.a. causality conditions; causality violations; Geodesically Complete; lorentzian geometry.
* Generic: Every timelike or null geodesic contains at least one point with non-zero 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 \(\cal I\)+, the closure of M′ ∩ J(\(\cal I\)+) 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.
@ General references: Manchak PhSc(11) [what is a physically reasonable spacetime?].
@ Conformally flat spacetimes: Pina & Tenenblat JGP(07); de Siqueira a1212-conf [?].
@ Totally vicious: Ikawa & Nakagawa JoG(88); Matori JMP(88); Kim & Kim JMP(93).
@ Topologically non-trivial: Lobo a1604-MG14 [multiply-connected spacetimes, rev]; > s.a. geons; wormholes.
@ Related topics: Markowitz GRG(82) [conformally hyperbolic]; Beem & Harris GRG(93) [generic]; Hall CQG(96) [decompositions into constant-type regions]; Ramos et al JMP(03) [double warped, invariant characterization]; Aké et al a1808 [globally hyperbolic, with timelike boundaries].
> 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.

Cylindrically Symmetric > s.a. axisymmetry [including higher dimensions]; 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: Buchdahl PR(59) [massless scalar field, Buchdahl solution]; Carot et al CQG(99)gq [def]; Arazi & Simeone GRG(00)gq; Barnes CQG(00)gq = gq/00-proc; 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]; Žofka & Bičák CQG(08)-a0712 [static, with cosmological constant]; Trendafilova & Fulling EJP(11)-a1101 [vacuum, static]; Giardino JModP(14)-a1403 [vacuum]; Bronnikov et al CQG(20)-a1901 [rev].
@ With massless scalar and cosmological constant: Momeni & Miraghaei IJMPA(09); Rezazadeh IJTP(11); Erices & Martínez PRD(15)-a1504 [stationary].
@ In other theories of gravity: Arazi & Simeone EPJP(11)-a1102 [Brans-Dicke theory]; Shamir & Raza ASS(15)-a1506 [f(R,T) gravity].

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]; 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]; Sung JGP(11) [non-existence of Einstein metrics on some classes of 4-manifolds]; Deshmukh JGP(11) [characterization]; Andersson & Moncrief JDG(11) [as attractors for the Einstein flow].
@ Stability: Barrow et al CQG(03) [with pfluid]; Carneiro & Tavakol PRD(09)-a0907 [in the presence of vacuum energy]; Grøn GRG(10); Barrow & Yamamoto PRD(12)-a1108 [as a non-LRS Bianchi type IX solution]; Kröncke AGAG(15)-a1311.
@ In other gravity theories, stability: Seahra & Böhmer PRD(09)-a0901 [instability in \(f(R)\) theories]; Böhmer & Lobo PRD(09) [in modified Gauss-Bonnet gravity]; Miritzis PRD(09) [in fourth-order gravity]; Goheer et al CQG(09); Böhmer et al MG12(12)-a1001; Canonico & Parisi PRD(10)-a1005 [in semiclassical lqc and Hořava-Lifshitz]; Atazadeh et al PLB(14) [in braneworld scenario]; > s.a. higher-order theories [hybrid theories]; hořava gravity.
@ Conformally Einstein: Gover m.DG/04-proc [almost]; Gover & Nurowski JGP(06)m.DG/04 [obstructions, n-dimensional].
@ Other generalizations: Ge et al a1508 [discrete]; Klemm & Ravera PRD(20)-a1811 [with torsion and non-metricity].

Stationary and Related Spacetimes > s.a. general relativity solutions with symmetries and with matter; Papapetrou Theorem.
* Ehlers group: A symmetry group of the vacuum Einstein equation for strictly stationary spacetimes.
* Stationary spacetime: One with a timelike Killing vector field, or with a one-parameter isometry group such that its orbits are everywhere timelike curves.
* Static spacetime: A stationary spacetime in which the timelike Killing vector field is hypersurface-orthogonal.
* Pseudostationary spacetime: A spacetime with a Killing vector field which is timelike at sufficiently large asymptotic distances.
@ Static: Bartnik & Tod CQG(06)gq/05 [and spatial 3-metrics]; Sánchez & Senovilla CQG(07) [global orthogonal decomposition]; Lafontaine JGP(09); Cederbaum a1210-proc [geometry, geometrostatics]; Ferrando & Sáez CQG(13)-a1302 [3-metrics that are spatial metrics of static vacuum solutions]; Figueiredo & Natário a2004 [duality with Riemannian manifolds]; > s.a. anti-de sitter spacetime; Ultrastatic Spacetimes.
@ Stationary: Tod CQG(17)-a1702 [characterizing initial data].
@ Related topics: Mars CQG(01)gq [Ehlers group]; > s.a. conformal transformations [extensions]; killing vector fields; spin coefficients.

Other Symmetries > s.a. general relativity solutions with symmetries; killing fields; lorentzian geometry; Maximally Symmetric Geometry.
@ 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]; Shabbir & Ramzan a1512 [non-static, proper curvature collineations].
@ 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 FLRW models.
@ Other spatial symmetries: Llosa & Carot CQG(10)-a0907 [two Killing vector fields, flat deformations]; > s.a. Self-Similarity; spherical symmetry [including Painlevé-Gullstrand form].

Other Types > s.a. coordinates [forms of line element]; examples of lie groups [spacetime groups].
* Generalized Lewis-Papapetrou: A metric of the form ds2 = −f (dt + ωi dxi)2 + f −1 gij dxi dx j ; > s.a. teleparallel gravity.
@ References: Clarke & Joshi CQG(88) [reflecting].
> Specific forms of the metric: see Gordon Ansatz; Kerr-Schild Metric; Misner Metric; Misner Space; Newman-Tamburino Metrics; W-Universe.
> Generalized geometries: see types of metrics [singular, distributional]; types of lorentzian geometries [including low-regularity]; types of singularities.

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