Bianchi Spacetime Models of Type I| Bianchi Spacetime Models of Type I |
In General
> s.a. bianchi I models with matter and in other theories.
$ Def: Defined by the structure constants
CABC
= 0 of the additive group \(\mathbb R\)3.
* Types of solutions: In vacuum
general relativity one gets the Kasner solution; With dust, one gets the
Heckmann-Schücking solution.
@ General references:
Heckmann & Schücking in(62);
Misner ApJ(68);
Schön pr(91) [new variables];
Bachmann & Schmidt PRD(00)gq/99 [quantum cosmology bifurcation];
Tsamparlis & Apostolopoulos JMP(00)gq [symmetries];
Khvedelidze & Mladenov PRD(02)gq [and 3-body Euler-Calogero-Sutherland model];
Shabbir & Khan MPLA(10) [classification];
Shabbir & Ali G&C(10) [proper projective collineations].
@ Solution space: Hervik CQG(00)gq [vacuum and dust];
Terzis & Christodoulakis CQG(12)-a1007 [entire solution space, Euclidean or Lorentzian].
@ Related topics: Cropp & Visser CQG(11)-a1008 [as blown-up neighborhoods of a timelike geodesic in any metric].
> Related geometrical topics:
see Collineations; coordinate systems
[geodesic lightcone coordinates].
> Related physical topics:
see cosmological models; gravitational
thermodynamics; observables; Silent Universe.
Kasner Solution > s.a. wave equation.
* Idea: Cosmological solution to the
vacuum Einstein's equation for the Bianchi I case, in which some dimensions expand
and some others contract in time (in 3+1 dimensions, one expands and two contract).
* Vacuum solution: It is given by
ds2 = −dt2
+ t2p1
dx2 + t2p2
dy2 + t2p3
dz2, with ∑i
pi = ∑i
pi2 = 1;
It can be parametrized by
p1 = −u (1+u+u2)−1 , p2 = (1+u) (1+u+u2)−1 , p3 = u (1+u) (1+u+u2)−1 .
* With fluid: In general relativity
with viscous fluid, it is not possible to obtain a model which satisfies the second
law of thermodynamics and the dominant energy condition; This can be done in other
theories, such as scalar-tensor gravity.
@ General references: Kasner TAMS(24);
Chodos & Detweiler PRD(80) [from Kaluza-Klein theory];
Harvey PRD(83);
Harvey GRG(90);
Gentle CQG(13)-a1208 [in Regge calculus].
@ Higher-dimensional:
Krori & Barua PLA(87) [9+1-dimensional];
Fabbri et al AACA(18)-a1812 [5D, with Dirac spinorss].
@ Generalized: Belinskii & Khalatnikov JETP(70) [mixmaster-like];
Gentle & Miller CQG(98)gq/97 [in Regge calculus];
Maceda et al EPJC(04)ht/03 [non-commutative];
Rasouli SPMS-a1405 [in Brans-Dicke theory];
> s.a. non-commutative gravity.
@ Related topics: Chicone et al PRD(11)-a1104 [cosmic jets in double-Kasner spacetimes];
Kofman JCAP(11) [perturbations].
Related Phenomena
> s.a. bel-robinson tensor; Boltzmann Equation;
electromagnetism in curved spacetime; light.
@ Isotropization: Fay CQG(01)gq/03 [scalar-tensor + massive scalar],
CQG(02)gq/03 [scalar-tensor + scalar + fluid];
Bronnikov et al G&C(02) [with spinor, vector and scalar fields];
Fliche et al IJMPD(03);
Fay & Luminet CQG(04)gq/03 [+ scalar];
Fay GRG(05)gq [+ non-minimal scalar];
Bronnikov et al IJTP(09) [+ electromagnetic + spinor field];
Rybakov et al IJTP(11)-a1006 [scalar field with non-linear potential];
Nungesser CQG(10) [collisionless matter, small anisotropy];
> s.a. bianchi I in other theories [f(R) gravity];
sources of gravitational radiation.
@ Perturbations: Banach JMP(99),
JMP(99);
Song NCB(00) [null geodesics];
Tsagas & Maartens CQG(00) [magnetized];
Pereira et al JCAP(07)-a0707;
Wilson & Dyer GRG(09) [planar];
Di Gioia & Montani EPJC(19)-a1807 [with a uniform magnetic field];
Agulló et al PRD(20)-a2003 [Hamiltonian theory],
a2006 [computational algorithm];
Boldrin & Małkiewicz a2105 [Hamiltonian formalism].
@ Phenomenology: Schücker et al MNRAS(14)-a1405,
a1601 [Hubble diagram of supernovae and anisotropy];
> s.a. gravitational radiation.
Quantization > see bianchi I quantum cosmology; semiclassical general relativity.
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