Types
of Spacetimes| 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 d
2.
@ 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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oct 2009