Cosmological solutions of the Einstein-Friedmann equations ...

Cosmology
With a 2 1
K
= 3
Λ
; a > 0 the mapping reads:
X = r sin θ sin ϕ
Y = r sin θ cos ϕ
Z = r cos θ
W = a 1 − r2
a2 cosh ct
a
T = a 1 − r2
a2 sinh ct
a
W = a
T = a
r 2
r 2
a2 − 1 sinh ct
a
a 2 − 1 cosh ct
a
⎫
⎪⎬
r < a
⎪⎭
⎫
⎪⎬
r > a .
⎪⎭
Note: z = constant is the hyperboloid: (cT) 2 − W2 = constant. Homogeneity and
Isotropy follow from the symmetry: S − 4 is invariant under 5-dimensional Lorentz
Λ r.
transformations. The force is linear repulsive F ∝ 1
3
c○ 2009, F. Jegerlehner ≪❘ Lect. 7 ❘≫ 484