6.2 The direction towards the singularity

In the study of the direction towards the singularity, it is convenient to change the time coordinate to t = e−τ, so that the singularity corresponds to τ → ∞. The metric then becomes
(τ−λ)āˆ•2 −2τ 2 2 −τ P 2 P P 2 −P 2 gT3 = e (− e dτ + dšœƒ ) + e [e dσ + 2e Qd σdδ + (e Q + e )d δ ]. (15 )
Here, τ ∈ ā„ and (šœƒ,σ,δ ) are coordinates on T3. Einstein’s vacuum equations take the form
−2τ 2P 2 −2τ 2 Pττ − e P šœƒšœƒ − e (Q τ − e Q šœƒ) = 0, (16 ) Q ττ − e−2τQ šœƒšœƒ + 2(P τQ τ − e −2τPšœƒQ šœƒ) = 0, (17 )
and
2 −2τ 2 2P 2 − 2τ 2 λ τ = Pτ + e P šœƒ + e (Qτ + e Q šœƒ), (18 ) λšœƒ = 2(PšœƒPτ + e2PQ šœƒQ τ). (19 )

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