The Mixmaster Model as a Cosmological Framework and Aspects of its Quantum Dynamics

20 Pages·0.31 MB·English

by  Giovanni Imponente

The Mixmaster Model as a Cosmological Framework and Aspects of its Quantum
Dynamics
Giovanni Imponente 1,2 ∗ and Giovanni Montani 1,3 †
1ICRA—International Center for Relativistic Astrophysics, Piazza A.Moro 5 00185 Rome, Italy
2Universit`a degli Studi di Napoli “Federico II” and INFN—Sezione di Napoli
3Physics Department, University of Rome “La Sapienza”
should characterize the very early evolution as soon as
Abstract: Thispaperprovidesareviewofsomerecentis-
we make allowance for small perturbations responsible
suesontheMixmasterdynamicsconcerningthefeaturesofits
for the actual clumpyness.
stochasticity. Afteradescriptionofthegeometrical structure
Furthermore the SCM in its original formulation con-
characterizing the homogeneous cosmological models in the
tains non-trivial internal inconsistencies which require
Bianchi classification and the Belinsky-Khalatnikov-Lifshitz
2 to be explained through appropriate modifications of
piecewise representation of the types VIII and IX oscillatory
0 the underlying theory. Among this undesired theoreti-
0 regime, wefacethequestionregardingthetimecovarianceof
cal facts take particular importance the so called “hori-
2 the resulting chaos as viewed in terms of continuous Misner-
zonparadox”and“flatnessproblem”to whichshouldbe
Chitr´e like variables. Finally we show how in the statistical
n addedthe absenceofanappropriatemodelforthe large-
mechanicsframeworktheMixmasterchaosraisesassemiclas-
a scalestructuresformationabletoreproducetheobserved
J sical limit of thequantum dynamicsin thePlanckian era.
distribution of matter in the actual universe [51,50,52].
1
All these considerations make clear the deep interest
3 PACS number(s): 04.20.Jb, 98.80.Dr
in studying more general classes of solutions of the Ein-

1 stein’s equations in order to individualize dynamical be-
v haviors which could constitute a more suitable frame-
2 work than the simple FRW model for the construction
0 I. INTRODUCTION of a completely self-consistent cosmology, with particu-
1 larreferencetotheuniverseevolutioninproximityofthe
1 As well-known the Standard Cosmological Model initial “Big Bang” [54,34,53].
0
(SCM) finds its theoretical basis in terms of a homo- In this paper (Sections I and II) we will discuss one of
2
0 geneous and isotropic universe obtained as high sym- the moststudied cosmologicaldynamicalmodels inview
/ metry solution of the Einstein’s equations, the so called ofits possibleimplicationsinthe historyofouruniverse,
c Friedmann-Robertson-Walkermodel(FRW).Suchrepre- the so called Mixmaster model [6] [7,19]. It corresponds
q
- sentation of our actual universe possesses a clear degree to the asymptoticevolutiontowardthe cosmologicalsin-
r of reliability due to its good general agreement with re- gularity of the type models VIII and IX in the famous
g
spect to the observed phenomenology (in particular the Bianchi classification [15]. Another important theoret-
:
v strong isotropy of the Cosmic Background Radiation as ical property characterizing such model is that it is an
Xi well as the consistency between the predictions on the appropriateprototypeofthe behaviorofthegeneralcos-
primordial nucleosynthesis of the light elements and the mologicalsolutionoftheEinstein’sequationsinthesame
r
a experimentallyobservedabundances),neverthelessthere asymptoticregiontotheinitialsingularity. Indeedthedi-
are some important general aspects to be taken into ac- rectextensiontotheinhomogeneouscaseoftheideasand
count. the formalism required by the treatment of the homoge-
In first place none theoretical principle led us to ex- neous one, extension based on the dynamical decoupling
clude that in the very early phases of its evolution the characterizingnearthe singularitythe differentpoints of
universehadbeencharacterizedbyahigherdegreeofin- the space, leads to derive in natural way the asymptotic
homogeneityandanisotropyandonlyinalaterstageun- evolution to the “Big Bang” of a generic inhomogeneous
derwentanisotropizationprocessasnaturalconsequence cosmological model which constitutes surely one of the
of its dynamics and/or by the action of some physical most important results until now obtained in Relativis-
mechanism in classical as well as quantum regime (to tic Cosmology [7,19].
which the causality notion plays a crucial role); indeed ThedescriptionoftheMixmasterModelcantakeplace
the instability of the FRW solution toward the cosmo- in terms of two different, but indeed completely equiv-
logical singularity implies that a more general behavior alent approaches: one is based on the direct analysis of
theEinstein’sequations,asintheoriginalanalysisdueto
Belinski, Khalatnikov and Lifshitz (BKL) (Section III),
while the other corresponds to a Hamiltonian approach
∗E-mail address: [email protected] tothedynamics(SectionIV)whichhadbeenintroduced
†E-mail address: [email protected] first by Misner [6].
1
However not all anisotropic dynamics are compatible II. GEOMETRICAL STRUCTURE OF THE
with a satisfactory SCM but, as shown in the early Sev- BIANCHI MODELS
enties,undersuitableconditionssomecanberepresented
asaFRWmodelplusagravitationalwavespacket([12], A. Group Transformations
[13]).
AmongtheBianchiclassification,thetypesVIII1 andIX
A (pseudo) Riemannian space, given a metric g and a
appear as the most general ones: the former’s geometry
differentialstructure (M,g),is homogeneousif itis orbit
is invariantunder the SO(3) group,like the closedFRW
for an isometry groupas a group invariance for the met-
universe and its dynamics allows the line element to be
ric g. The groups of movements for the metric are said
decomposed as
to be isometry groups and for homogeneous spaces they
are transitive, in the sense that each point is equivalent
ds2 =ds 2 δ G(a)(b)dxidxk (1)
0 − (a)(b) ik under the action of the group and they are spatial sec-
tions of the space-time.
where ds denotes the line element of an isotropic uni-
0 We are interested in the Lie algebras for Killing vector
verse having positive constant curvature, G(a)(b) is a set
ik fields as generators of these groups of movements, in-
ofspatialtensors2 andδ (t)areamplitude functions,
(a)(b) tended as infinitesimal transformations generating rota-
resulting small sufficiently far from the singularity.
tions and translations.
Since Belinski, Khalatnikov and Lifshitz [7] derived the
Let’s consider a group of transformations
oscillatory regime characterizing the evolution near a
physicalsingularity,awideliteraturefacedovertheyears xµ x¯µ =fµ(x,a) (2)
this subject in order to understand the resulting chaos. →
Theresearchdevelopedoverallintwodifferent,butre- over a space M, where aa are r independent
{ }a=1,..,r
lated, directions: firstly, the dynamicalanalysisremoved variables parameterizing the group and a corresponds
0
thelimitsoftheBKLapproachduetoitsdiscretenature, to the identical transformation
providingsatisfactoryrepresentationsintermsofcontin-
uous variables (i.e. construction of an invariant measure fµ(x,a0)=xµ. (3)
for the system [21], [48]); secondly, a better characteri-
Let’s then consider also the infinitesimal transformation
zation of the Mixmaster chaos (in view of its properties
corresponding to a +δa, close to the identity,
of covariance) found non-trivial difficulties to apply the 0
standard chaos indicators to relativistic systems.
xµ x¯µ =fµ(x,a +δa)
The various approaches prevented, up to now, to say a → 0 ≈
definitivewordaboutthecovarianceofMixmasterchaos, fµ(x,a )+ ∂fµ (x,a )δaa (4)
though importantindications in favorof such covariance ≈ 0 ∂aa 0
(cid:18) (cid:19)
arise from [47], [60], [57] and [58,59], as above discussed =xµ
in detail. ≡ξaµ(x)
| {z }
A wide interest in understanding this intrinsic nature say | {z }
ofthechaoticMixmasterdynamicsanditsveryearlyap-
pearanceintheUniverseevolution,towhichwededicate xµ →x¯µ ≈xµ+ξaµ(x)δaa =(1+δaaξa)xµ, (5)
ouranalysis,leadto believe inthe existence ofarelation
where the r firstorder differentialoperators ξ arede-
with the quantum behavior the system performs during a
fined as ξ =ξµ ∂ in correspondence with{the}r vecto-
the Planckian era. In fact the last aim of this paper a a∂xµ
rial fields with components ξµ . These are the Killing
(Section V) is to give a precise meaning to such relation { a}
generating vector fields.
by constructing the semiclassical limit of a Scho¨edinger
The coordinate transformation can be definitely written
approach to the canonical quantization of the Arnowitt-
as
Deser-Misner (ADM) dynamics.
x¯µ (1+δaaξ ) eδaaξaxµ (6)
a
≈ ≈
and in a finite form
1AlltheconsiderationswewilldevelopforthetypeIXapply
also to the VIII one since, close to the singularity, they have x¯µ x¯µ =eθaξaxµ (7)
→
thesame morphology.
2These tensors satisfy theequations where θa are r new parameters for the group.
{ }
The vectorial generator fields form a Lie algebra, say a
G(aik)(b;)l;l=−(n2−3)G(aik)(b), G(a;)k(b)k =0, G(ai)(b)i =0, realr-dimensionalvectorialspacewith basis ξa , closed
{ }
withrespectto commutation,inordertohavethe repre-
in which the Laplacian is referred to the geometry of the
sentation
sphere of unit radius.
[ξ ,ξ ] ξ ξ ξ ξ =Cc ξ (8)
a b ≡ a b− b a ab c
2
whereCc arethestructureconstantsfortheLiealgebra. Inanycaseoneneedstoconsideronlyonerepresentative
ab
It is then natural to extend this formalism to the choice groupforeachclassofequivalenceoftheLiegroups. Re-
of e asbasisfortheLiealgebrag overagroupGwith ferring to three dimensions (only the spatial sector), the
a
{ }
Bianchiclassification[1]determinesdefinitelyallsymme-
[ea,eb]=Ccabec; (9) triesfortridimensionalhomogeneousspaces,analogously
to the curvature (k = 1,0,+1) which distinguishes ho-
then let’s define the symmetric quantity −
mogeneous and isotropic spaces (FRW).
γ =Cc Cd =γ (10)
ab ad bc ba
B. Einstein’s Equations in the Synchronous Gauge
and an internal product over the Lie algebra
γ e e =γ(e ,e ) , γ Xae ,Ybe =γ XaYb. Let’s us specify the general scheme outlined above in
ab a b a b a b ab
≡ ·
the specific approach of Luigi Bianchi in 1897 [1] and
(cid:0) (cid:1) (11)
independently appliedtocosmologybyBelinski,Khalat-
nikov and Lifshitz in 1969 [7].
The orbit of x is
In the cosmologicalapproach we consider a synchronous
f (x)= f (x) a G (12) reference system, so that there is an unique time t syn-
G a
{ | ∈ } chronizedalloverthespace. Thiscanbeobtainedrequir-
asthesetofallpointsthatcanbeachievedfromxunder ing the metric tensor g to have the peculiar diagonal
ab
the groupof transformations. The isotropygroupin x is form such as
Gx = a G fa(x)=x , (13) g00 =1 g0α =0, α=1,...,3 (16)
{ ∈ | }
i.e. the subgroup of G which leaves x fixed. If x ,x in order to write the line element
1 2
are on the same orbit, then G and G are conjugate
x1 x2 ds2 =N2(t)dt2 dl2, (17)
subgroups of G and then isomorphic. If now Gx = a0 −
{ }
and fG(x) = M, G is diffeomorphic to M and the two whereN(t)isthelapsefunction(inthissectionweadopt
spaces are identified. If g is a metric over M invariant the synchronous gauge N 1). Writing3
underG,thenitisdefinitivelygivenbytheinnerproduct ≡
of the invariant vectorial fields of basis ea. dl2 =γαβ(t,xγ)dxαdxβ, (18)
Thegroupsofnon-Abeliantransformations,asdefined
t is the synchronous time and the tridimensional ten-
by (9), represent geometrically homogeneous spaces in
sor γ describes the metric spatial sector. The iden-
three dimensions and give the section spatially homoge- αβ
tity g 0 ensures the complete equivalence in the
neous of the spatially homogeneous space-times. 0α ≡
three spatial directions, while the unitarity of g is
Given a basis e of the Lie algebra for the tridimen- 00
{ a} always valid, provided an appropriate rescaling of the
sional Lie group, with structure constants Cc , at any
ab time coordinate. In this reference system the time lines
time the spatialmetric is givenby the spatially constant
(xγ = const.) are the geodesics of the four dimensional
inner products
space-time
e e =g (t) a,b=1,...,3 (14)
a· b ab dui +Γi ukul =Γi =0, (i,k,l=0,...,3) (19)
which are six functions of the time variable only. This ds kl 00
permitstorewritetheEinsteinequationsasordinarydif- being ui = dxi the tangent four vector to the world line
ferential equations, eventually associated to constraints ds
xγ = const. with components u0 = 1, uα = 0 and or-
functions of t, necessary to describe the matter behavior
thogonal to the hypersurfaces t=const.4.
of the universe.
Infourdimensionsoneobtainshomogeneousspace-times:
given a set of structure constants Cα over a basis e
βγ { α}
of the four dimensional Lie algebra, the ten constants
3All over this section Latin indexes run over 0,...,3 while
g =e e α,β =1,...,4 (15) Greek indexes1,...,3.
αβ α β
4This geometrical construction is always available but not
determineunivocallythesignatureoftheLorentzmetric. univocally defined: a coordinate transformation of thetype
As a consequence, Einstein’s equations reduce to a sys-
t′ =t
temofalgebraicequationsforgαβ andCαβγ,whichcould xα′ =xα′ xβ (20)
in principle not be solvable for each transformations’ (cid:26)
group, in view of the fact that for general formulations doesn’t affect the time coordin(cid:0)ate(cid:1)and defines the passage
with matter fields there are more constants to take ac- from a synchronous reference system to another one, fully
count of. equivalent.
3
Underthischoiceandthemetric(17),themixedcompo- inversematrix is ηab, being ηacη =δa. Thenfollow the
cb b
nents of the Einstein’s equations write properties
R0 = 1 ∂ κα 1κβκα =8πG T0 1T (21) η(a)(b)e(ia) =e(b)i (27)
0 −2∂t α− 4 α β 0 − 2
(cid:18) (cid:19) η(a)(b)e =e(b) (28)
1 (a)i i
Rα0 = 2 κβα;β −κββ;α =8πGTα0 (22) e(a)ie(ja) =gij . (29)
(cid:16) 1 ∂(cid:17) 1
Rβ = Pβ √γκβ =8πG Tβ δαT (23) For a generic vector (A ) or tensorial (T ) field the
α − α − 2√γ∂t α α − 2 β j ij
(cid:18) (cid:19) tetradic components are, in general
(cid:0) (cid:1)
being A =e Aj =ej A , (30)
(a) (a)j (a) j
καβ = ∂γαβ, γ γαβ , (24) A(a) =η(a)(b)A(b) =e(ja)Aj =e(a)jAj, (31)
∂t ≡| |
Ai =ei A(a) =e(a)iA (32)
(a) (a)
T themomentumenergytensorforthesystemandP
αβ αβ
the tridimensionalRiccitensor obtainedfromthe metric and, respectively,
γ 5.
αβ
In (23) the spatial and temporal derivatives are split, T(a)(b) =ei(a)ej(b)Tij =ei(a)Ti(b), (33)
while from (21) one obtains the property for the met-
T =e(a)e(b)T =e(a)T . (34)
ric determinant to be zero for a certain time (Landau- ij i j (a)(b) i (a)j
Raichoudhury theorem)6, nevertheless the singularity
The tetradic indexes can be raised or lowered with the
withrespecttothetemporalparameterisaphysicalone. useofthetensorsη andη(a)(b),whilethecontraction
(a)(b)
gives then a result independent of the indexes nature.
By (26) and (27) one obtains also
C. Tetradic Representation of the Metric Tensor
g =e e(a) =η e(a)e(b) (35)
ik (a)i k ab i k
The choice of a tetradic basis of four linearly indepen-
dent vector fields (depending from the system symme- so that the line element becomes
tries) permits to project all quantities in a very useful
way to obtain new simpler equations to be satisfied. ds2 =ηab e(ia)dxi ek(b)dxk . (36)
Consider on each point of the space-time the basis of
(cid:16) (cid:17)(cid:16) (cid:17)
fourlinearlyindependentcontravariantvectorsei ,(a= The choice of η permits to split the tetradic basis in
(a) ab
1,...,4), where the bracketed index is the tetradic one, one temporalandthree spatialvectors. Nevertheless the
while i is tensorial,andthe covariantsete =g ek , expressions dx(a) = e(a)dxi are not, in general, exact
(a) i ik (a) i
being g the metric tensor. Define also the reciprocal differentials of functions of the coordinates.
ik
set e(a)ei = δa (orthogonality condition). It is easy to
i (b) b
check the validity of
D. Directional Derivative
e(a)ek =δk. (25)
i (a) i
The contravariant set e of tangent vectors leads to
(a)
The definition is complete with the natural covariantderivative definition
∂
ei(a)e(b) i =ηab, (26) e(a) =ei(a)∂xi (37)
where ηab is a symmetric matrix with signature (+ sothatthederivativeofagenericscalarfieldφalongthe
−
)(withtheorthonormalityof(25));thecorresponding direction a is
−−
∂φ
φ =ei =ei φ . (38)
,(a) (a)∂xi (a) ,i
5The indexes of these two tensors are raised and lowered The general extension of such definition is
using such reduced metric.
6Thisfictitioussingularitydisappearswithachangeofrefer- A =ei ∂ A =ei ∂ ej A =
encesystemandisrelatedtotheintersectionofthegeodesics (a),(b) (b)∂xi (a) (b)∂xi (a) j
belonging to any arbitrary set with the envelope surfaces,
=ei ej A =ei ej A +A ek (39)
say to the procedure used to build a synchronous reference (b)∇∂i (a) j (b) (a) j;i k (a);i
system. h i h i
in order to rewrite
4
A =ej A ei +e ei ek A(c). (40) R =R em ei ek el =
(a),(b) (a) j;i (b) (a)k;i (b) (c) (a)(b)(c)(d) mikl (a) (b) (c) (d)
= γ +γ +
(a)(b)(c),(d) (a)(b)(d),(c)
Let’s introduce the Ricci’s rotation coefficients γ −
abc
+γ γ (f) γ (f) +
(a)(b)(f) (c) (d) (d) (c)
γ =e ei ek (41) −
abc (a)i;k (b) (c) +γ hγ (f) + i
(f)(a)(c) (b) (d)
and their linear combinations
γ γ (f) . (51)
(f)(a)(d) (b) (c)
−
λ =γ γ = e e ei ek =
abc abc− acb (a)i;k− (a)k;i (b) (c) In view of the re-expression of the interesting quantities
= e(a)i,k−e(a)k(cid:0),i ei(b)ek(c) (cid:1) (42) isnubtseerqmusenotflythienRtiecrcmi’ssrooftatthieonstcrouecffitucrieenctsonγs(taa)(nbt)(sc,)iatnids
in which is (cid:0)has been used t(cid:1)he identity easy to see the great simplification in the formulas, pro-
vided a space for which exists a set of such constants
A A = ∂Ai ∂Ak . (43) C(a) , like the analysis made by Bianchi for the homo-
i;k− k;i ∂xk − ∂xi (b)(c)
geneous spaces leads to [1].
Expression(42),inwhichtheregularderivativesaresub- The length element before the transformation is
stituted by the covariantones, is invertible
dl2 =γ x1,x2,x3 dxαdxβ (52)
αβ
1
γabc = (λabc+λbca λcab) . (44) that, under a general(cid:0)change of(cid:1)coordinates, transforms
2 −
to
From the identity
dl2 =γ x′1,x′2,x′3 dx′αdx′β (53)
αβ
0=η = e ek (45)
(a)(b),i (a)k (b) where the functiona(cid:0)l form γ (cid:1)has to be the same un-
;i αβ
h i der the homogeneity constraint. Such requirement, in a
one gets the symmetry properties uniform non Euclidean space, leads to invariance of the
threeindependentdifferentialformsin(36)whicharenot
γ = γ , λ = λ . (46)
abc bac abc acb exact differential of any function of the coordinates and,
− −
in tetradic form, are
Nowtheformalismisreadytofindthevaluesofthestruc-
tureconstantswhichleavethemetricinvariantunderthe e(a)dxα. (54)
α
homogeneity constraint.
The basis e permits to express the Lie parentheses as The spatial invariant line element rewrites
(a)
e(a),e(b) =C(c()a)(b)e(c) (47) dl2 =ηab e(αa)dxα eβ(b)dxβ (55)
(cid:16) (cid:17)(cid:16) (cid:17)
where the coeffi(cid:2)cients C(cid:3)(c) are the 24 (in four dimen- so that the metric tensor becomes
(a)(b)
sional space) structure constants for the group of trans- γ =η e(a)e(b), (56)
αβ ab α β
formations, antisymmetric with respect to the lower in-
dexes; it is easy to obtain the explicit relation maintaining for η the definition (26), as a function of
ab
the time variable only.
(c) (c) (c)
C =γ γ . (48) The symmetry properties for the space determine the
(a)(b) (b)(a)− (a)(b)
specific choice of the basis vectors which, in general, are
not orthogonal and consequently the metric η is not
ab
diagonal.
E. Ricci and Bianchi Identities In pure specific case the relations between the three vec-
tors are
By the Riemann tensor Rm , the Bianchi identity be-
ikl 1
comes e = e(2) e(3)
(1) v ∧
1h i
Ai;k;l−Ai;l;k =AmRmikl (49) e(2) = v e(3)∧e(1) (57)
for a generic four vector Ai and, applied to the tetradic e = 1he(1) e(2)i
basis, (3) v ∧
h i
e e =em R . (50) where v is the product
(a)i;k;l− (a)i;l;k (a) mikl
Projecting this expression on the basis itself one obtains v =|e(αa) |= e(1)· e(2)∧e(3) (58)
(cid:16) h i(cid:17)
5
and where it is natural to interpret e and e(a) as the Multiplying (65) by eγ gives the uniformity conditions
(a) (c)
Cartesian vectors of components eα and e(a) respec- over the space
(a) α
tively. ∂eγ ∂eγ
The metric tensor determinant (56) takes the value eα (b) eβ (a) =Cc eγ . (66)
(a) ∂xα − (b) ∂xβ ab (c)
γ =ηv2 (59)
The expression on the left hand side corresponds to the
definition of λc (42), then constants.
being η the determinant of the matrix ηab. ab
The antisymmetry property holds, see (47) or (48), with
The space invariance as in (53) is equivalent to the in-
respect to the lower indexes
variance of (54), then
eα(a)(x)dxα =e(αa)(x′)dx′α (60) Ccab =−Ccba. (67)
Such relations can be rewritten in a compact form in
(a)
where e on both sides of the equation are the same
α terms of the linear operators
functions of the old and new coordinates respectively.
Making some algebra one gets ∂
X =eα (68)
a (a)∂xα
∂x′β
=eβ (x′)e(a)(x) , (61)
∂xα (a) α so that (66) becomes
which is a system of differential equations to determine [X ,X ] X X X X =Cc X , (69)
the functions x′β(x) on the given basis. The integrabil- a b ≡ a b− b a ab c
ity of this system requires the satisfaction of Schwartz’s and the homogeneity is expressed as Jacobi identity
conditions
[[X ,X ],X ]+[[X ,X ],X ]+[[X ,X ],X ]=0
a b c b c a c a b
∂2x′β ∂2x′β
= , (62) (70)
∂xα∂xγ ∂xγ∂xα
which, in terms of the structure constants, reads
and by explicit calculations leads to
∂eβ (x′) ∂eβ (x′) CfabCdcf +CfbcCdaf +CfcaCdbf =0. (71)
(a) eδ (x′) (b) eδ (x′) e(b)(x)e(a)(x)=
∂x′δ (b) − ∂x′δ (a) γ α Bydualtransformationonecangetthetwoindexesstruc-
h ∂e(a)(x) ∂e(a)(x) i ture constants, more convenient for calculations
=eβ (x′) γ α . (63)
(a) " ∂xα − ∂xγ # Cc =ε Cdc (72)
ab abd
Usingthepropertiesofthetetradicbasisandmultiplying where ε = εabc is the Levi-Civita unitary antisym-
abc
both sides of (63) by eα (x)eγ (x)e(f)(x′) and some metric tensor (ε = +1). The commutation rules (69)
(d) (c) β 123
algebra the expression on the left hand side becomes expressedinthenewconstantsacquirethecompactform
∂eβ (x′) ∂eβ (x′) εabcXbXc =CadXd. (73)
e(f)(x′) (d) eδ (x′) (c) eδ (x′) =
β " ∂x′δ (c) − ∂x′δ (d) # Antisymmetryisimpliedinthe definition(72),while the
∂e(f)(x′) ∂e(f)(x′) Jacobi identity (71) becomes
=eβ (x′)eδ (x′) β δ . (64)
(c) (d) " ∂x′δ − ∂x′γ # εbcdCcdCba =0 (74)
Analogously one obtains for the righthand side an iden- and (66) for the set (72) is equivalent to the vectorial
tical expression but different only for being function of form
x. Being the transformation x x′ arbitrary, both ex-
1
pressions have to be equal to th→e same constant, giving Cab = e(a)rot e(b). (75)
−v
the group constant of structure7
Any linear transformation with constant coefficients
∂e(c) ∂e(c)
α β eα eβ =Cc . (65) e =Abe(b) (76)
∂xβ − ∂xα ! (a) (b) ab (a) a
showsthe nonunivocalchoice ofthe three referencevec-
torsinthedifferentialforms(54)andwithrespecttosuch
transformationsη and Cab behave like tensors. Condi-
ab
7For theindexes of Cc theparentheses are unimportant. tion (74) is the only one to be satisfied by the structure
ab
6
constants, considering only non equivalent combinations III. PIECEWISE REPRESENTATION OF THE
with respect to the transformation (76). MIXMASTER
The classification of non equivalent homogeneous spaces
reduces to the determination of all non equivalent com- A. Field Equations
binations of the constants Cab.
Imposing condition (74) one gets the relations
In a synchronous reference system, the metric for a
homogeneous model writes
[X ,X ] = aX +n X
1 2 2 3 3
−
[X2,X3] =n1X1 (77) ds2 =dt2−ηab(t)e(αa)(xγ)eβ(b)(xγ)dxαdxβ (78)
[X ,X ] =n X +aX
3 1 2 2 3
where the reference vectors e(a) are determined through
α
where a and (n1,n2,n3) are constants related to the (66), once specified the structure constants. The matrix
structure constants. Upon reduction of all non equiv- η (t) describes the temporal evolution of the tridimen-
ab
alentsetsunder rescalingallpossible uniformspacescan sional geometry, to be derived from the Einstein’s field
be summarized following the Bianchi classification as in equations which reduce to an ordinary differential sys-
the following table. tem, involving functions of the t only, provided the pro-
jection of all spatial part of vectors and tensors over the
tetradic basis chosen, using
Type a n1 n2 n3 R =R eα eβ
I 0 0 0 0 (a)(b) αβ (a) (b)
II 0 1 0 0 R0(a) =R0αeα(a)
VII 0 1 1 0 T =T e(a)e(b) (79)
VI 0 1 -1 0 αβ (a)(b) α β
IX 0 1 1 1 T =T e(a)
α0 (a)0 α
VIII 0 1 1 -1
u(a) =uαe(a).
α
V 1 0 0 0
IV 1 0 0 1 Homogeneityreflectsoverallscalarquantitiespreventing
VII a 0 1 1 the presence of any spatial gradient, incompatible with
III (a=1) the problem symmetry.
a 0 1 -1
VI (a6=1) ) The matrix ηab is the projection over the basis of the
spatial metric γ and the role of η and ηab to raise
αβ ab
TABLEI. Bianchiclassification–Nonequivalentstructure and lower the indexes is clear.
constants The projection of the field equations (21)-(23) over the
tetrad gives
1 1
R0 = κ˙(a) κ(b)κ(a) (80)
0 −2 (a)− 4 (a) (b)
1
R0 = κ(c) Cb δbCd (81)
(a) −2 (b) ca− a dc
(a) 1 ∂(cid:0) (b) (cid:1)(a)
R = √ηκ P (82)
(b) −2√η∂t (a) − (b)
(cid:16) (cid:17)
where
κ =η˙
(a)(b) (a)(b)
κ(b) =η˙ η(c)(b) (83)
(a) (a)(c)
and the dot is the derivative with respect to t and the
tridimensional Ricci tensor projected
P =η P(c) (84)
(a)(b) (b)(c) (a)
in terms of the structure constants becomes
1
P = CcdC +CcdC +
(a)(b) − 2 b cda b dca
1(cid:16)
C cdC +Cc C d+Cc C d . (85)
− 2 b acd cd ab cd ba
(cid:17)
7
Moreoverit is not necessarythe explicit formof the vec- γ =t2p1n(1)n(1)+t2p2n(2)n(2)+t2p3n(3)n(3) (95)
ab a b a b a b
tor basis as functions of the coordinate for . This task is
relatedtothesymmetrydegreeofthemodelsconsidered. where the constant coefficients for the powers of t can
be reduced to unity, once rescaled the coordinates. If
thetetradvectorsareparalleltothecoordinateaxes,say
(x,y,z), metric reduces to
B. Kasner Solution
ds2 =dt2 t2p1dx2 t2p2dy2 t2p3dz2. (96)
− − −
The simplest homogeneous cosmological model corre-
Constants p ,p ,p are three arbitrary numbers, called
1 2 3
sponds to the Bianchi type I, whose structure constants
Kasner indexes, which have to satisfy the conditions
identically vanish, implying alsothe Ricci tensorcompo-
nents to be zero so that p +p +p =1 (97)
1 2 3
p2+p2+p2 =1, (98)
ea =δa 1 2 3
α α = P =0. (86)
Cacb ≡0) ⇒ ab asadirectconsequenceof(94)and(93). Exceptthecases
(0,0,1)and( 1,2,2), Kasnerindexes haveto be differ-
Under this conditions and in empty space, equations ent and one o−f3th3em3can acquire negative value. Given
(80)-(82) reduce to the system the order
1
κ˙a+ κbκa =0 (87) p1 <p2 <p3, (99)
a 2 a b
1 ∂ the corresponding variation interval is
√γκb =0. (88)
√γ∂t a
1
(cid:0) (cid:1) p 0
1
By (88) one gets the first integral − 3 ≤ ≤
2
0 p (100)
√γκb =2λb =const.. (89) ≤ 2 ≤ 3
a a
2
p 1.
The contraction of the a and b indexes gets 3 ≤ 3 ≤
γ˙ 2 Thesenumberscanalsoberepresentedintheparametric
κa = = λa, (90)
a γ √γ a form
u
and then p (u) = −
1 1+u+u2
γ =Gt2, G=const.. (91) 1+u
p (u) = (101)
2 1+u+u2
Without loss of generality, upon a coordinates rescaling, u(1+u)
p (u) =
put such constant equal to one and then also 3 1+u+u2
λa =1. (92) wheretheparameteruvariesintheinterval1 u<+ .
a ≤ ∞
Whenu<1,the parameterizationcanbe reducedtothe
Substituting (89) in (87) one finds the relation for the same variability interval for p ,p ,p , holding
1 2 3
constants λb
a
1
λabλba =1. (93) p1 u =p1(u)
(cid:18) (cid:19)
1
Lowering index b in (89) gives a system of ordinary dif- p =p (u) (102)
2 3
ferential equations with respect to γ u
ab (cid:18) (cid:19)
1
2 p =p (u).
γ˙ = λcλ . (94) 3 u 2
ab t a cb (cid:18) (cid:19)
Asfunctionsofu,theparametersp (u)andp (u)mono-
The set of λc can be considered as a matrix for a linear 1 3
a tonically increase, while p (u) monotonically decreases.
transformation and by an appropriate change of coordi- 2
Thismetriccorrespondstoaflatspace,evenanisotropic,
nates (x1,x2,x3) is reducible to a diagonal form. Given
where the volume grows proportionally with increasing
p ,p ,p as the corresponding matrix eigenvalues, taken
1 2 3
t and distances along (y,z) axes increase while along
real and different, in correspondence of the normalized
eigenvectors n(1),n(2),n(3), the solution of (94) is x decrease. The time t = 0 is the singular point for
the solution and such singularity is not avoidable under
8
any reference system change, while the invariant quan- initial conditions at a given time, but the general solu-
tities of the curvature tensor diverge except the case tion satisfies arbitrary initial conditions, then the given
p = p = 0, p = 1 in which the metric is reducible perturbation cannot affect the form of the solution.
1 2 3
to the Galilean form once posed the parameterization Thestructureconstantssetsconsideredare(see table)
tsinhx3 =ξ Bianchi VIII: C11 =C22 =1, C33 = 1
−
tcoshx3 =τ. (103) Bianchi IX: C11 =C22 =1=C33 =1. (106)
The metric (96) is the exact solution for the Einstein’s Let’s consider the case of Bianchi IX. If the matrix ηab
equations in empty space but, close the a singular point, is diagonal, the components R0 of the Ricci tensor in
(a)
say small t, it represents an approximate solution. (80)-(82) vanish identically in the synchronous reference
Ageneralizedsolutioncoincidesonlywithanapproxima- system and, by (85), the same holds for the off diagonal
tion of the metric, in the sense that dominant terms of components of P . The remaining Einstein’s equa-
(a)(b)
such metric as powers of t have analogous form to (96). tions give for the functions a(t),b(t),c(t) the system
In general, given a synchronous reference system, the
•
metric canbe expressedin the form(17), where the spa- (a˙bc) = 1 b2 c2 2 a4
tial line element dl is abc 2a2b2c2 − −
dl2 = a2lαlβ +b2mαmβ +c2nαnβ dxαdxβ (104) ab˙c • = 1 h(cid:0)a2 c2(cid:1)2 b4i (107)
(cid:16)abc(cid:17) 2a2b2c2 − −
once posed(cid:0) (cid:1) (abc˙)• = 1 h(cid:0)a2 b2(cid:1)2 c4i
a=tpl abc 2a2b2c2 − −
h(cid:0) (cid:1) i
b=tpm (105) and
c=tpn a¨ ¨b c¨
+ + =0 (108)
and the three tridimensional vectors l,m,n redefine the a b c
directions along which the spatial lengths vary following
dependingonthetimevariableonlyand()• = d. Equa-
the power laws (105). Such quantities are functions of dt
tions (107)-(108) are exact and valid also far from the
the spatial coordinates. In view of the fact that expo-
singularity in t=0. Introducing the quantities
nents (105) are all different, the spatial metric (104) is
always anisotropic. a=eα
An eventual presence of matter doesn’t affect the gener-
b=eβ (109)
ality of these conclusions and can be introduced in the
metric (104)-(105) via four coordinates functions to de- c=eγ
termine the initial distribution of matter and the three
and the temporal variable τ
initial velocity components. The behavior of matter in
the vicinity of a singular point is determined by equa-
dt=abc dτ (110)
tions of motion in a given gravitational field, following
classical hydrodynamics.
one obtains the simplified form
2α = b2 c2 2 a4
ττ
C. Oscillatory Regime − −
2βττ =(cid:0)a2 c2(cid:1)2 b4 (111)
− −
Differently from the Kasner solution, in all other ho- 2γττ =(cid:0)a2 b2(cid:1)2 c4
mogeneous models the projection of the tridimensional − −
1
Ricci tensor has components different from zero and the (α+β+γ) =α(cid:0) β +α(cid:1) γ +β γ (112)
2 ττ τ τ τ τ τ τ
complexification induced prevents an analytical descrip-
tionofthesolution,exceptfortheBianchitypeIImodel. wherethe indexτ referstothe derivativewithrespectto
In the following is described the asymptotic behavior of τ. The system (111), by (112), admits the first integral
the homogeneous models in the vicinity of a singular
point. α β +α γ +β γ =
τ τ τ τ τ τ
The most general and interesting case relies in the 1
= a4+b4+c4 2a2b2 2a2c2 2b2c2 (113)
BianchitypesVIII andIXmodels(Mixmaster)[6],while
4 − − −
others can be considered as simplifications. (cid:16) (cid:17)
The general solution is, by definition, stable. A pertur- containing only first derivatives. The Kasner regime
bation to the system is equivalent to a change in the (105) is a solution of (111) if all right hand side expres-
sions can be neglected. Such a situation cannot hold
9
indefinitely (for t 0), because there is always an in- where b and c are integration constants. For the so-
0 0
→
creasing term. Suppose the case that, for a certain time lutions (120), in the limit τ , the asymptotic form
→ ∞
interval, some expressions in (111) are negligible (Kas- coincides with (116). For τ , say toward the sin-
→ −∞
nerian regime): for example, the negative exponent cor- gularity, the asymptotic forms become
responds to the function a(t), so that p = p , and the
l 1
perturbation to the given regime is due to terms as a4, a exp[Λp1τ]
∼
meanwhile other terms decrease. Keeping the dominant b exp[Λ(p +2p )τ]
2 1
∼
terms in (111), one reduces to the approximate
c exp[Λ(p +2p )τ] (121)
3 1
∼
1 t exp[Λ(1+2p )τ]
αττ = e4α ∼ 1
−2
1 and writing a,b,cas functions of t one gets againa Kas-
βττ =γττ = e4α. (114) ner behavior, a new Kasner epoch,
2
The solution for those equations describes the evolution a tp′l
of the metric, given the initial condition (105). Without b∼tp′m (122)
loss of generality, once put ∼
c tp′n
p =p ∼
l 1 abc=Λ′t
p =p (115)
m 2
p =p being
n 3
so that p
p′ = | 1 |
a tp1 l 1−2|p1 |
∼ 2 p p
b tp2 (116) p′ = | 1 |− 2 (123)
∼ m −1 2 p
c tp3 − | 1 |
∼ p 2 p
p′ = 3− | 1 |,
one obtains n 1 2 p
1
− | |
abc=Λt
together with
1
τ = lnt+const. (117)
Λ Λ′ =(1 2 p )Λ. (124)
1
− | |
where Λ is a constant while initial conditions for (114)
The perturbation causes the transition from a Kasner
are
epoch to another in such a way that the negative power
α =p of t passes from the direction l to m, i.e. if initially p
τ 1 l
is negative, in the new solution one gets p ′ < 0. The
β =p (118) m
τ 2 initial perturbation given by terms such e4α is reduced
γ =p ,
τ 3 substantially to zero. For the following evolution it will
and finally become the dominant term in another right hand side
term and follow a completely analogous analysis.
α=Λp
1 By the parameterization (101) and the property (102),
β =Λp (119) the passages from a Kasner epoch to another re-express
2
γ =Λp . in a suggestive way such that if
3
The first of (114) has the same form as the equation p =p (u)
l 1
of the unidimensional motion for a point particle in an p =p (u) (125)
m 2
exponential potential barrier, where α has the role of a
p =p (u)
coordinate. Bythisanalogy,theinitialKasnerianregime n 3
correspondstoafreemotionwithconstantvelocityατ = then
p . Afterthebounceonthebarriertheparticlefollowsits
1
free motion with the same speed but with opposite sign p′ =p (u 1)
l 2 −
ατ = p1. Using initial conditions (119), the system p′ =p (u 1) (126)
(114) in−tegrates as m 1 −
p′ =p (u 1)
n 3 −
2 p Λ
a2 = | 1 | and is called BKL map.
cosh(2 p Λτ)
1
| | Duringthistransition,thefunctiona(t)getsamaximum
b2 =b02exp[2Λ(p2 p1 )τ]cosh(2 p1 Λτ) (120) value, while b(t) a minima. After that b starts increas-
−| | | |
c2 =c 2exp[2Λ(p p )τ]cosh(2 p Λτ) ing, a decreases and c(t) holds its monotonic decrease.
0 3 1 1
−| | | |
10