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].

> __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].

@ __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].

**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]; > 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 Specific Forms** > s.a. coordinates [forms
of line element]; types of metrics [including singular, distributional]
and types of singularities.

* __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]; > Gordon Ansatz; Kerr-Schild Metric; Misner
Metric; Misner
Space; Newman-Tamburino Metrics; W-Universe.

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