32 Kinematics and dynamics of an expanding universe
t∼= const χ∼= constFig. 1.9.where+∞> ̄t>−∞and+∞>χ> ̄ −∞.Expressingzin terms oft ̄,χ ̄, one
finds that only the half of the hyperboloid located atx+z≥0 is covered by these
“flat” coordinates (Figure 1.10). The metric becomes
ds^2 =d ̄t^2 −H−^2 exp (2H ̄t)dχ ̄^2. (1.98)The relation between the different coordinate systems in the regions where they
overlap can be obtained by comparing (1.93), (1.95) and (1.97):
cosh(Ht) cosχ=cosh(Ht ̃)=cosh(H ̄t)−^12 exp(H ̄t)χ ̄^2 ,
cosh(Ht) sinχ=sinh(H ̃t) sinh ̃χ=exp(H ̄t)χ. ̄(1.99)
De Sitter spacetime via analytical continuation (three-dimensional case)Since a
de Sitter universe is a spacetime of constantpositivecurvature with Lorentzian
signature, it can be obtained by analytical continuation of a metric describing a
positive curvature space with Euclidean signature. To see how analytical continu-
ation changes the signature of the metric let us consider (1.39) describing a closed
universe(k=+ 1 ).After the change of variables,
a→H−^1 ,χ→Hτ,θ→χ,φ→θ,