Figure 4

Figure 4: Conformal diagram of the critical solution matched to an asymptotically flat one (RC: regular centre, S: singularity, SSH: self-similarity horizon). Curved lines are lines of constant coordinate τ, while converging straight lines are lines of constant coordinate x. Let the initial data on the Cauchy surface CS be those for the exact critical solution out to the 2-sphere R, and let these data be smoothly extended to some data that are asymptotically flat, so that the future null infinity ℐ + exists. To the past of the matching surface MS the solution coincides with the critical solution. The spacetime cannot be uniquely continued beyond the Cauchy horizon CH. The redshift from point A to point B is finite by self-similarity, and the redshift from B to C is finite by asymptotic flatness.