130	5 Cosmology and General Relativity

Fig. 5.4 The cosmic billiard mechanism envisages solutions of Einstein equations that include a series of Kasner epochs following each other as a result of a bounce on the walls of a billiard table. In higher dimensional supergravities the billiard table turns out to be the Weyl chamber of the duality Lie algebra pertaining to the considered model

Fig. 5.4). This provides a possible new paradigm for the interpretation of the extradimensions that occur in superstring based supergravity models. Not necessarily compact, such extra dimensions might be effectively small because depressed by decreasing (even exponentially decreasing) scale factors.

In this perspective the billiard mechanism implies that the effective dimensions of space-time might change with time. While entering a new Kasner phase, the Universe might acquire new dimensions which were previously contracting and now they might start expanding, while old expanding dimensions might contract and progressively disappear. The first idea of such a scenario was put forward by the Russian physicists Belinsky, Khalatnikov and Lifshitz in [3–6].

5.3.2A Toy Example of Cosmic Billiard with a Bianchi II Space-Time

Here we do not present the very rich systematics of supergravity and gravity billiards, for which we refer the interested reader to some comprehensive research papers and lecture notes [7]. Our goal is just that of illustrating the main conception of the billiard mechanism by means of a simple toy model that we can realize in d = 4 space-time dimensions. The chosen toy model corresponds also to a Bianchi type of homogeneous but not isotropic universe which helps us to emphasize the role of isotropy in our subsequent discussion of the Standard Cosmological Model.

5.3.2.1 A Ricci Flat Bianchi II Metric

To begin with we study a particular cosmological metric which is Ricci flat and therefore corresponds to an empty Universe that, nonetheless, expands in some of

Ω1 = −dz +

5.3 Homogeneity Without Isotropy: What Might Happen

131

its space-like directions. The Bianchi type II exact solutions presented in this and in the following subsection were derived a few years ago by this author with his collaborators Trigiante and Rulik in [8]. The metric is of the following form:

ds(d2

4) = −A(t) dt2 + Λ(t) Ω22 + Ω32 + Δ(t)Ω12

(5.3.20)

where the three 1-forms Ωi are explicitly given by:

ωr2

dθ

4

Ω2 = r cos θ dθ + sin θ dr (5.3.21)

Ω3 = − cos θ dr + r sin θ dθ

and satisfy the following Cartan Maurer equations:

	ω
dΩ1	=		Ω2	Ω3
		2
dΩ2	= 0			(5.3.22)
dΩ3	= 0			(5.3.23)

This means that the constant time sections of the space-time (5.3.20) are 3- dimensional homogeneous spaces, namely copies of a three dimensional group manifold Gω whose corresponding Lie algebra is the following non-semisimple one

[Ti , Tj ] = tijk Tk
t231 =	ω	; all other components of tijk vanish	(5.3.24)
	4

As it is the case for any group manifold, there exist on Gω two mutually commuting sets of vector fields that separately satisfy the Lie algebra of the group, namely the generators of left translations and the generators of right translations. Let us

agree that the 1-forms (5.3.21) are left invariant. Then the triplet of vector fields

−→

that generate the left translations ki will be such that they satisfy the Lie algebra (5.3.24) and the Lie derivative of the Ωi along them vanishes.

[	−→, −→						t	−→		(5.3.25)
	k	i		k	j ] =			k
							ij
			i			=	0			(5.3.26)
		−→		Ωj
	k

The explicit form of such vector fields is the following one:

−→1

= ∂z

k

∂

−→2

= √ω

∂r

+ √ωr

∂θ +

2

∂z

2

∂

2

∂

√ω

∂

k

sin θ

cos θ

r cos θ

(5.3.27)