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 *K*^{a}
*K*^{b}
*K*_{[c}
*R*_{d]ab[e}
*K*_{f ]} (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 *a*^{2}
+ *e*^{2} > *m*^{2}.

@ __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 *R*_{ab}
= *c* *g*_{ab}, 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 TS^{n+1}];
Gibbons et al CQG(04)ht
[5D, on S^{3}-bundle over S^{2}];
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 d*s*^{2}
= −*f* (d*t* + *ω*_{i}
d*x*^{i})^{2}
+ *f*^{ −1} *g*_{ij}
d*x*^{i} d*x*^{ 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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