Bianchi
Classification of Cosmological Models| Bianchi
Classification of Cosmological Models |
In General > s.a. Homogeneity;
Isotropic Metric.
* Idea: It is a classification
of most spatially homogeneous anisotropic cosmologies; What Bianchi really
did was to classify the three-dimensional
Lie algebras, but the Lie algebra can be taken to be that of the isometry group,
and the two things become equivalent; The original scheme
has been later modified.
$ Def: Spacetimes with
a 3D group G of isometries acting freely and (simply)
transitively on spacelike hypersurfaces.
* Motivation: The vacuum
or perfect fluid models give simple equations, which however are non-linear
enough to study anisotropy, singularities, chaotic
dynamics,
horizon structure, gravitational waves, non-Newtonian Weyl curvature, etc.
* Classification: If
the Lie algebra is generated by Killing vector fields 
i (i =
1, 2, 3), with commutators [i,
j]
= Cijk k,
choose a triad of left-invariant vector fields Li such
that
Cijk =
ijl nlk
+ 
jk vi – ik vj
,
where nij = diag(n1,n2,n3), ni =
1, 0, and vi =
(0,1; 0,1; 0,1), with vi nij
= 0; Types are classified by the combinations of values of
ni and vi (class
A has vi = 0, class B has vi
0); The dual forms 
i
are also left-invariant.
* Killing vector fields:
They generate the isometries, so they are right-invariant
and satisfy
[i, j]a =
Ckij ka
, [, 
]a = Cabc b c .
* Left-invariant forms:
There are three 1-forms 
ia,
such that qab = gij ia jb and
[a ib] = –
Ccab ic.
* Metric: It is often parametrized in an ADM-like form by a volume
element and a shear matrix 
, such that
ds2 = –N2(t)
dt2
+ gij(t)
ij , gij
= exp{–2
(t)}
(exp{2(t)})ij ;
the simplest case is for types I, II, VIII, IX, in which g or are
diagonalizable; With some loss of generality,
gij = exp{–2(t)}
diag(a(t), b(t), 1/ab) , ij
= diag(++
31/2 –, +– 31/2 –,
–2+
) .
Special Cases and Models > s.a. bianchi
I, bianchi IX, and
other models; scalar-tensor theories.
@ Class/type A: Schücking et al GRG(03) [diagram]; Apostolopoulos CQG(03)gq [self-similar].
@ Class/type B: Apostolopoulos CQG(05)gq/04
[self-similar].
@ Compact: Koike et al JMP(94)
[topologies]; Tanimoto et al JMP(97)gq/96, JMP(97)gq;
Coley & Goliath PRD(00)gq [fluid
+ scalar];
Kodama PTP(02)gq/01 [fluid];
Torre CQG(04)gq [weakly
locally homogeneous].
References > s.a. geodesics;
minisuperspace quantum cosmology; quasilocal
energy; types
of singularities.
@ Early references: Bianchi MMSI(1897),
translation GRG(01); Bianchi 18; Misner PR(69)
[ADM].
@ Reviews: in Misner et al 73; Ryan & Shepley 75; MacCallum in(79);
Jantzen in(84); Ellis GRG(06).
@ EMW variables: Ellis & MacCallum CMP(69);
MacCallum & Ellis CMP(70);
MacCallum CMP(71);
Wainwright in(88); Wainwright & Hsu CQG(89); Wainwright & Ma in(92);
Hewitt & Wainwright CQG(93).
@ Ashtekar variables: Kodama PTP(88);
Ashtekar & Pullin in(90);
Ashtekar & Samuel
CQG(91);
Manojlovic & Mikovic CQG(93);
Obregón et al PRD(93)gq;
González & Tate
CQG(95)gq/94 [I
and II]; Calzetta & Thibeault gq/97/CQG
[I, II, IX].
@ Symmetries and reduction: Coussaert & Henneaux CQG(93)gq;
Schirmer CQG(95);
Capozziello et al IJMPD(97)
[Noether symmetries and general relativity]; Christodoulakis et al CMP(02)gq/01 [invariant
characterization]; Apostolopoulos CQG(05)
[new approach].
@ Hamiltonian / Lagrangian: Ryan & Waller gq/97 [class
B]; Tanimoto et al JMP(97)gq [compact];
Pons & Shepley PRD(98)gq.
@ Dynamics, general: Bonilla CQG(98);
Byland & Scialom PRD(98)gq [I,
II, KS]; Billyard gq/99-PhD;
Szydlowski & Krawiec
IJMPA(00)
[constraint solved]; Aguiar & Crawford PRD(00)gq [I
+ III, dust and 
];
Barrow & Hervik
CQG(02)gq [Weyl
curvature invariant]; Gambini & Pullin CQG(03)
[discretized]; Fay gq/05-in
[+ massive scalar, ADM]; Goheer et al CQG(07)-a0710
[in f(R) theories].
@ Stability: Barrow & Sonoda PRP(86)
[several V, VI, VII models]; Zotov gq/99.
@ Isotropization: Cervantes-Cota & Chauvet PRD(99)gq/98 [I-V-IX, induced
gravity]; Fay CQG(03)
[class A], gq/05-in
[scalar-tensor]; > s.a. types I, VII, IX.
@ In Einstein-Yang-Mills theory: Donets et al PRD(99)
[N = 2 supersymmetric].
@ Other theories: Savaria gq/97 [non-symmetric
gravity]; Bonneau CQG(98)
[Weyl]; Querella gq/99-PhD
[higher-derivative]; Barrow & Hervik PRD(06)gq [I
+ II, quadratic theories]; Cognola & Zerbini a0802 [generalized gravity theories]; > s.a. modified
electrodynamics, non-commutative physics.
@ Phenomenology: Coley & Lim CQG(07)
[bounds on shear]; > s.a. cmb anisotropy.
@ Related topics: Estabrook et al JMP(68);
Siklos CMP(78)
[horizons and whimper singularities]; Rosquist & Jantzen
PRP(88);
Fujiwara et al CQG(93)gq;
Di Pietro & Demaret IJMPD(99)gq [duality];
Rendall & Uggla CQG(00)gq [Einstein-Vlasov];
De Ritis et al NCB(01) [horizons];
Doliwa et al JGP(04)
[discretization]; Hervik ap/05,
Palle ap/05 [re
vorticity, > s.a. cmb].
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send feedback and suggestions to bombelli at olemiss.edu – modified 9
aug
2009