5.3 Homogeneity Without Isotropy: What Might Happen |
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In the case of symmetric spaces Cabe = 0 formula (5.2.100) simplifies to:
Rabcd = − |
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2λ2 Cabi Cicd |
(5.2.101) |
5.2.3.5 For Spheres and Pseudo-Spheres
In order to illustrate the structures presented in the previous section we consider the explicit example of the spheres and pseudo-spheres. Applying the outlined procedure to this case we immediately get:
Ea = − |
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dya |
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κ |
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ωab = 2 |
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This means that for spheres and pseudo-spheres the Riemann tensor is proportional to an antisymmetrized Kronecker delta:
Rabcd = |
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(5.2.103) |
5.3 Homogeneity Without Isotropy: What Might Happen
Having prepared the stage with our discussion of coset manifolds, among which group manifolds are a particular case, we enter the main issue of mathematical cosmology by utilizing the above developed techniques in order to construct spacetime metrics that display homogeneity with the possible addition of full or partial isotropy. The goal is that of understanding how Einstein equations turn into differential equations for the free functions fi (t) of the time variable t that parameterize such homogeneous metrics and solve them if possible. The behavior of the solutions encodes the possible scenarios of cosmic evolutions.
It is quite important to understand that the two features advocated by the Cosmological Principle, namely homogeneity and isotropy are completely independent. For this reason, in this section, we present solutions of the Einstein equations based on homogeneous but not isotropic metrics. The corresponding cosmic evolution is very different from the overall expansion motivated by Hubble Law. Without isotropy the space-like sections of the Universe not only expand or contract, but also continuously deform during time-evolution.
5.3.1 Bianchi Spaces and Kasner Metrics
A very simple way to realize a four-dimensional cosmological metric which is homogeneous without any a priori enforcement of isotropy relies on the use of the
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5 Cosmology and General Relativity |
Maurer Cartan forms Ωi of a three-dimensional Lie group G3 satisfying the equation:
dΩi = tij k Ωj Ωk |
(5.3.1) |
where tij k are the structure constants of the corresponding Lie algebra G3. Assuming that the Ωi are, for instance, left-invariant we have that:
kI Ωi = 0; I = 1, 2, 3 |
(5.3.2) |
where the vector fields kI are the infinitesimal generators of the left translations. Introducing the ansatz:
dsG2 = −dt2 + Aij (t)Ωi Ωj |
(5.3.3) |
where Aij (t) is a time-dependent symmetric positive definite matrix we obtain a metric which is Lorentzian and admits the three vector fields kI as space-like translational Killing vectors. The group G3 has a transitive action on the constant time sections of such a space-time, which therefore realizes a homogeneous but a priori not isotropic cosmology.
One can insert the ansatz (5.3.3) into the Einstein equations and look for solutions with various types of matters.
The very interesting point is that in 1898 the Italian geometer Luigi Bianchi, the same who is responsible for Bianchi identities, succeeded in obtaining a complete classification of all possible three-dimensional real Lie algebras [1]. The key argument utilized by Bianchi which exploits the peculiar features of three dimensions is
the following. Given the structure constants ti |
one can define the following new |
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Ak = tiik |
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(5.3.5) |
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As a consequence of its definition the matrix M i |
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Mi is symmetric. In terms of |
these new objects, the vector Ak and the matrix M , the Jacobi identities reduce to |
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the very simple condition: |
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Hence the classification of all three-dimensional Lie algebras was reduced to the classification of solutions of (5.3.6). By means of rotations in the basis of generators the vector Ak can be oriented in a conventional direction, say the first, and the matrix M k can be diagonalized. Using this liberty the form taken by the Maurer Cartan equations of the Bianchi algebras is the following one:
dΩ1 |
= λ1 |
Ω |
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Ω |
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Ω |
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dΩ3 = λ3Ω1 Ω2 − aΩ3 Ω1 |
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5.3 Homogeneity Without Isotropy: What Might Happen |
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Table 5.1 The classification by Bianchi of three-dimensional Lie algebras |
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Bianchi type |
a |
λ1 |
λ2 |
λ3 |
Identification |
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I |
0 |
0 |
0 |
0 |
R3 |
II |
0 |
1 |
0 |
0 |
Heisenberg algebra |
III |
1 |
0 |
1 |
−1 |
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IV |
1 |
0 |
0 |
0 |
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V |
1 |
0 |
0 |
0 |
Solv(so(1, 3)/so(3)) |
VI0 |
0 |
1 |
−1 |
0 |
iso(1, 1) |
VIa |
a |
0 |
1 |
−1 |
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VII0 |
0 |
1 |
1 |
0 |
iso(2) |
VIIa |
a |
0 |
1 |
1 |
so(1, 2) sl(2, R) |
VIII |
0 |
1 |
1 |
−1 |
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IX |
0 |
1 |
1 |
1 |
so(3) |
The algebras VIa and VIIa are actually two continuous families of solvable non-isomorphic Lie algebras, distinguished by the value of the parameter a. Some of the Bianchi algebras can be identified with other well known, simple or solvable Lie algebras. Bianchi IX and Bianchi VIII are simple and correspond to the two possible real sections, respectively compact and non-compact, of the unique complex Lie algebra A1. Bianchi VII0 is the Lie algebra of the Euclidian group of the plane E2 iso(2), while Bianchi VI0 is the Poincaré Lie algebra in two dimensions iso(1, 1). Bianchi I is just the translation algebra R3, while Bianchi II is the Heisenberg algebra in two dimension. Finally according to a general theorem every non-compact simple coset manifold G/H, where H is a maximally compact subgroup, is metrically equivalent to a solvable group manifold exp Solv where the corresponding solvable Lie algebra Solv can be constructed from the Lie algebra G, according to a well defined procedure. Bianchi V is the solvable Lie algebra associated with the pseudo-sphere SO(1, 3)/SO(3).
and the various solutions are classified by specifying the four numbers λ1,2,3 and a. In this way Bianchi obtained the list of eleven algebras displayed in Table 5.1. An interesting class of solutions of Einstein equations is obtained by using the Abelian algebras of Bianchi type I.
5.3.1.1 Bianchi Type I and Kasner Metrics
Within the general class of Bianchi I metrics that can be written as follows:
dsBianchi2 I = −dt2 + Aij (t) dxi dxj |
(5.3.8) |
we can consider the subclass of Kasner metrics [2], defined below:
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2p |
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5 Cosmology and General Relativity |
where pi are real exponents. The Vielbein description of the metric (5.3.9) is the following one:
E0 = dt; Ei = tpi dxi
which leads to the following spin connection
ω0i = 1 pi Ei ; ωij = 0
t
and curvature 2-form: |
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yielding the following Ricci tensor (in flat indices):
(5.3.10)
(5.3.11)
(5.3.12)
Ric00 = |
2t2 |
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pi |
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It follows that the Kasner metric is a solution of vacuum Einstein equations, namely it is Ricci flat whenever the exponent pi satisfies the following two algebraic equations:
3 |
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pj = 1 |
(5.3.14) |
j =1 |
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Geometrically the locus singled out by (5.3.14) is the intersection of a two-sphere with a plane and corresponds to a curve in three-dimensions that is displayed in Fig. 5.3. A parametric solution of equations is given below:
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Equation (5.3.15) provides just one branch of the overall solution. The other branches are obtained by applying the permutation group S3 to it and altogether they fill the curve presented in Fig. 5.3.
Any point on this curve {p1, p2, p3} yields a possible vacuum solution of Einstein equations that is named a Kasner epoch. In such Kasner epochs the destiny of the various space-dimensions is very different: some contracts, other expands, since
5.3 Homogeneity Without Isotropy: What Might Happen |
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Fig. 5.3 The curve of Kasner exponents {px , py , pz} corresponding to Ricci flat metrics
the exponents pi have typically different signs. For instance a nice rational solution of the Kasner constraints is provided by:
{p1, p2, p3} = |
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Let us now consider metrics of the following type: |
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dsK2 = −dt2 + ai2(t)Ωi2 |
(5.3.17) |
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i=1 |
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where Ωi are the Maurer Cartan forms of a Bianchi Lie algebra not necessarily of type I. One can draw a mechanical analogy by identifying:
hi (t) = log ai (t) |
(5.3.18) |
with the coordinates of a fictitious ball that is moving in a three-dimensional space with velocity:2
d |
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vi (t) = dt log ai (t) |
(5.3.19) |
Kasner epochs correspond to constant velocity trajectories.
A very interesting feature arising while discussing homogeneous non isotropic solutions of Einstein equations is that of cosmic billiards. These latter are exact solutions of matter coupled higher dimensional gravity where a succession of different Kasner epochs are glued together, one after the other, in a smooth but sharp way (see
2In higher dimensional gravity theories the ball moves in n-dimensions.