6.1 Expanding direction

The starting point is the metric (2View Equation). For metrics of this form, Einstein’s vacuum equations are equivalent to
Ptt + 1Pt − P 𝜃𝜃 − e2P(Q2t − Q2𝜃) = 0, (11 ) t 1- Qtt + tQt − Q𝜃𝜃 + 2(PtQt − P 𝜃Q𝜃) = 0, (12 )
2 2 2P 2 2 λt = t[P t + P𝜃 + e (Q t + Q𝜃)], (13 ) λ𝜃 = 2t(P𝜃Pt + e2PQ 𝜃Qt). (14 )
Since the equations for P and Q do not depend on λ, it is possible to solve Equations (11View Equation) – (12View Equation) first and then to calculate λ using Equation (14View Equation). In order for this procedure to be meaningful, it is necessary to choose the initial data for P and Q in such a way that the integral of the right-hand side of Equation (14View Equation) vanishes. However, assuming that has been done, λ is determined up to a constant given a solution to Equations (11View Equation) – (12View Equation).
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