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].

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