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.