Choptuik Spacetime as an Eigenvalue Problem

By fine-tuning generic Cauchy data, critical phenomena have recently been discovered in the black-hole/no-black-hole “phase transition” of various gravitating systems. For the spherisymmetric scalar field system, we find the “critical” spacetime separating the two phases by demanding discrete scale invariance, analyticity, and an additional reflection-type symmetry. The resulting nonlinear hyperbolic boundary value problem, with the rescaling factor $\Delta$ as the eigenvalue, is solved numerically by relaxation. We find $\Delta =3.4439\mathrm{}\pm 0.0004$.