Curvature blow up, polarized T3-case

7.4 Curvature blow up, polarized T³-case

From the point of view of proving strong cosmic censorship, the main reason for wanting to derive expansions of the form of Equation (35

) is that they can be used to prove curvature blow up. It turns out that criteria for curvature blow up and the absence thereof can be formulated in terms of $v$ . Consider a past inextendible causal curve $γ$ . Due to the form of the metric (2

), it is clear that the $𝜃$ -coordinate of this curve has to converge in the direction towards the singularity. Call the limiting value $𝜃0$ . Then, if

$v2(𝜃0) ⁄= 1$ ,
$2 v (𝜃0) = 1$ but $∂𝜃v(𝜃0) ⁄= 0$ , or
$2 v (𝜃0) = 1$ , $∂ 𝜃v (𝜃0) = 0$ , but $2 ∂𝜃v(𝜃0) ⁄= 0$ ,

the Kretschmann scalar is unbounded along $γ$ in the direction of the singularity. Otherwise, it is bounded. This result, as well as quantitative estimates for the rate at which the curvature tends to infinity, is contained in [50 , Theorem IV.1, p. 105].

7.4 Curvature blow up, polarized T3-case

7.4 Curvature blow up, polarized T³-case