Bianchi Spacetime Models of Type IX| Bianchi Spacetime Models of Type IX |
In General
> s.a. chaos in bianchi models; gravitational
instanton; minisuperspace quantum cosmology.
$ Def: In the Lie-algebra
classification, nij
= (+1, +1, +1), vi =
0; The group G is (the simply connected covering group of) SO(3).
* History: The
Russian school (BKL) started studying Bianchi IX in 1962, hoping to
understand the behavior of the metric near a generic singularity
(introduced map for u); Misner started in 1966, but using
earlier work on Taub-NUT, hoping to understand anisotropy dissipation
(introduced Hamiltonian and potential).
* Special cases: \(l_1 = l_2 = l_3
= {1\over2}\,R\), FLRW models; Kasner solution; Taub-NUT solution; Diagonal models,
the metric has gab
= diag(l1, l2,
l3), with li
functions of time; The diagonal vacuum model is also called Mixmaster universe.
* Geometry: 3V
= 16π2 l1
l2 l3,
ω1 = cosψ dθ + sinψ sinθ dφ , ω2 = sinψ dθ − cosψ sinθ dφ , ω3 = dψ + cosθ dφ .
* Evolution: Well approximated by a sequence of Bianchi I (Kasner) epochs; Each one is characterized by the value of a parameter u, which gives rise to an approximate discrete dynamics, the Gauss map
un+1 = (un − [un])−1 ;
In the Hamiltonian approach, each epoch is the time between two bounces
off the potential; At each bounce, two scale factors switch between
expansion and contraction, while the third one keeps contracting; An
era is a set of epochs with the same two factors switching
behavior, i.e., bouncing off the same pair of walls.
* Better approximation: & Garfinkle.
References
> s.a. gravitational energy-momentum; types
of spacetime singularities; Taub-NUT Solution [early work].
@ General: Harvey PRD(83) [new solutions];
Montani et al IJMPA(08)-a0712 [classical and quantum, review].
@ ADM approach: Misner ApJ(68),
PR(69),
PRL(69);
in Misner et al 73;
in Ryan & Shepley 75;
Imponente & Montani gq/02-in,
IJMPD(02).
@ BKL approach:
Belinskii et al AiP(70),
JETP(71),
AiP(82);
Manojlović & Miković JMP(00)mp [Painlevé III];
Imponente & Montani JKPS(03)gq/02.
@ Other approach:
Creighton & Hobill in(94) [Ellis-MacCallum-Wainwright];
Gogilidze et al G&C(97) [Hamiltonian, non-diagonal].
@ Dynamics near the singularity:
Czuchry & Piechocki PRD(13)-a1202 [non-diagonal models];
Czuchry et al a1409 [comparing diagonal and non-diagonal cases];
Parnovsky & Piechocki a1605;
Kiefer et al EPJC(18)-a1807 [simplified dynamics, numerical simulations].
@ Other dynamics:
Llibre & Valls JMP(05),
JMP(06) [Darboux first integrals];
Buzzi et al JPA(07);
Starkov PLA(11) [compact
invariant sets; no periodic, homoclinic, or heteroclinic orbits in the zero-level set of the Hamiltonian];
Dimakis et al PRD(19)-a1809 [Liouville integrability];
Barrow a2006 [synchronisation of oscillations];
> s.a. chaos in bianchi models;
early-universe models.
@ Self-dual:
Tod PLA(94);
Chakravarty & Ablowitz PRL(96);
Maszczyk CQG(96).
@ Self-similar: Apostolopoulos & Tsamparlis GRG(03)gq.
@ Isotropization: Guzman IJTP(96);
Bergamini et al PRD(97)gq/96 [inflation];
Cervantes-Cota & Chauvet PRD(99)gq/98 [induced gravity];
Kirillov & Montani PRD(02)gq [and inflation];
Battisti et al a0903-proc [semiclassical mechanism].
@ With matter: Waller PRD(84) [electromagnetism];
Banerjee et al ASS(90) [viscous fluid];
Toporensky & Ustiansky gq/99,
Fay & Lehner GRG(05)gq [massive scalar];
Farajollahi & Ravanpak IJTP(09)-a1001 [massless scalar];
Saha G&C(13)-a1107 [restrictions on the components of the energy-momentum tensor];
Pavluchenko PRD(16)-a1607 [Einstein-Skyrme];
Saha CJP(18)-a1705 [spinor field];
> s.a. types of spacetimes [instability].
@ Collapse:
Lin & Wald PRD(90) [recollapse];
Charters a1106
[vacuum, proof of collapse conjecture].
@ In Hořava-Lifshitz gravity:
Myung et al PRD(10)-a0911 [chaotic and non-chaotic solutions],
JHEP(10)-a1001;
Bakas et al CQG(10)-a0911 [and chaos];
Misonoh et al PRD(11)-a1104.
@ Other theories:
Belinskii et al PLB(78),
in Cotsakis 90 [Euclidean];
Barrow & Dąbrowski PRD(98)ht/97 [low-energy string theory];
Garcia de Andrade & Monerat ap/01/C&G [with torsion];
Halpern GRG(03)gq/02 [5D];
van den Hoogen et al PRD(03)gq/02 [brane];
Bergshoeff et al CQG(03)ht [supergravity];
Maceda et al PRD(08) [non-commutative];
> s.a. modified uncertainty relations;
non-commutative gravity.
@ Related topics: Chitre PRD(72) [wave equation];
Hu PRD(73) [Klein-Gordon fields];
King PRD(91);
in Misner in(94) [as geodesic motion];
Berger et al CQG(97)gq/96,
gq/97-conf [other algorithms];
Cotsakis et al PRD(98)gq/97 [adiabatic invariants and catastrophes];
Barguine et al PRD(01)
[with cosmological constant, homoclinic structure];
Battisti & Montani a0903-proc [gup approach];
Shabbir et al G&C(10) [proper curvature collineations].
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 2 jun 2020