By Woei Chet Lim; University of Waterloo. Dept. of Applied Mathematics
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Additional resources for The dynamics of inhomogeneous cosmologies
1)Hypersurface-orthogonal This case is specified by requiring that the G2 admits a hypersurfaceorthogonal KVF ξ , ξ[a;b ξc] = 0. 1) The following proposition provides a convenient way to check for the existence of a hypersurface-orthogonal KVF. 29 CHAPTER 3. 1. Consider a G2 cosmology presented in a group-invariant orbit-aligned frame with e2 and e3 tangent to the G2 orbits. 45) satisfy σ ˜AB n ˜ AB = 0, n ˜ AB σ 1A σ 1B = 0, ∗ σ ˜AB σ 1A σ 1B = 0. 46) satisfy ǫAB qA σ1B = 0, ǫAB π1A σ1B = 0, ∗ π ˜AB σ 1A σ 1B = 0.
3. 53). Note that Bianchi VII cosmologies with generic G2 action necessarily have the most general tilt, as shown in the following proposition. 2. Consider a G2 -compatible SH cosmology presented in a group-invariant orbit-aligned frame with e2 and e3 tangent to the G2 orbits, CHAPTER 3. THE G2 AND SH HIERARCHIES 36 and e1 tangent to the G3 orbits. If det nAB > 0 and σ1A σ 1A = 0, then ǫAB vA σ1B = 0. Proof. Suppose ǫAB vA σ1B = 0. Contract (CC )A with ǫAB σ1B to obtain nC C σ1A σ 1A = 2˜ nAB σ 1A σ 1B .
46 CHAPTER 4. SH DYNAMICS seem so. 3. 2. 101) were introduced for the purpose of analyzing G2 -compatible SH cosmologies. e. ) rather than in the 1+1+2 form, so that we can impose a suitable spatial gauge condition. 24) NC C = 2N+ , N N× −N− and Σ12 = √ 3Σ3 , Σ13 = √ 3Σ2 . 85), and are given below: 47 CHAPTER 4. 34) where 2 2 q = 2Σ2 + 12 G−1 + [(3γ − 2)(1 − v ) + 2γv ]Ω − ΩΛ 2 Σ2+ Σ2− Σ2× 2 2 2 Σ = + + + Σ2 + Σ3 G± = 1 ± (γ − 1)v , v 2 = v12 + v22 + v32 . 37) Constraint equations: 0 = (CG ) = 1 − N−2 − N×2 − A2 − Σ2 − Ω − ΩΛ γΩ 0 = (CC )1 = 2AΣ+ + 2(Σ− N× − Σ× N− ) + v1 G+ √ √ √ γΩ v2 0 = (CC )2 = −(3A + 3N× )Σ3 − (N+ − 3N− )Σ2 + 3 G+ √ √ √ γΩ v3 .
The dynamics of inhomogeneous cosmologies by Woei Chet Lim; University of Waterloo. Dept. of Applied Mathematics